The Pythagorean Theorem - Educational Outreach

The PythagoreanTheoremCrown Jewel of Mathematics534John C. Sparks

The PythagoreanTheoremCrown Jewel of MathematicsBy John C. Sparks

The Pythagorean TheoremCrown Jewel of MathematicsCopyright © 2008John C. SparksAll rights reserved. No part of this book may be reproducedin any form—except for the inclusion of brief quotations in areview—without permission in writing from the author orpublisher. Front cover, Pythagorean Dreams, a compositemosaic of historical Pythagorean proofs. Back cover photo byCurtis SparksISBN: XXXXXXXXXFirst Published by Author House XXXXXLibrary of Congress Control Number XXXXXXXXPublished by AuthorHouse1663 Liberty Drive, Suite 200Bloomington, Indiana 47403(800)839-8640www.authorhouse.comProduced by Sparrow-Hawke †reasuresXenia, Ohio 45385Printed in the United States of America2

Conceptual Use of the Pythagorean Theorem byAncient Greeks to Estimate the DistanceFrom the Earth to the SunSignificanceThe wisp in my glass on a clear winter’s nightIs home for a billion wee glimmers of light,Each crystal itself one faraway dreamWith faraway worlds surrounding its gleam.And locked in the realm of each tiny sphereIs all that is met through an eye or an ear;Too, all that is felt by a hand or our love,For we are but whits in the sea seen above.Such scales immense make wonder aboundAnd make a lone knee touch the cold ground.For what is this man that he should be madeTo sing to The One Whose breath heavens laid?July 19994

Table of ContentsList of Tables and Figures 7PageList of Proofs and Developments 11Preface 131] Consider the Squares 172] Four Thousand Years of Discovery 272.1] Pythagoras and the First Proof 272.2] Euclid’s Wonderful Windmill 362.3] Liu Hui Packs the Squares 452.4] Kurrah Transforms the Bride’s Chair 492.5] Bhaskara Unleashes the Power of Algebra 532.6] Leonardo da Vinci’s Magnificent Symmetry 552.7] Legendre Exploits Embedded Similarity 582.8] Henry Perigal’s Tombstone 612.9] President Garfield’s Ingenious Trapezoid 662.10] Ohio and the Elusive Calculus Proof * 682.11] Shear, Shape, and Area 832.12] A Challenge for all Ages 863] Diamonds of the Same Mind 883.1] Extension to Similar Areas 883.2] Pythagorean Triples and Triangles 903.3] Inscribed Circle Theorem 963.4] Adding a Dimension 983.5] Pythagoras and the Three Means 1013.6] The Theorems of Heron, Pappus, 104Kurrah, Stewart3.7] The Five Pillars of Trigonometry 1163.8] Fermat’s Line in the Line 130* Section 2.10 requires knowledge of college-level calculusand be omitted without loss of continuity.5

Table of Contents…continuedPage4] Pearls of Fun and Wonder 1364.1] Sam Lloyd’s Triangular Lake 1364.2] Pythagorean Magic Squares 1414.3] Earth, Moon, Sun, and Stars 1444.4] Phi, PI, and Spirals 157Epilogue: The Crown and the Jewels 166Appendices 169A] Greek Alphabet 170B] Mathematical Symbols 171C] Geometric Foundations 172D] References 177Topical Index 1796

List of Tables and FiguresTablesNumber and TitlePage2.1: Prior to Pythagoras 282.2; Three Euclidean Metrics 692.3: Categories of Pythagorean Proof 863.1: A Sampling of Similar Areas 893.2: Pythagorean Triples with c < 100 923.3: Equal-Area Pythagorean Triangles 953.4: Equal-Perimeter Pythagorean Triangles 953.5: Select Pythagorean Radii 973.6: Select Pythagorean Quartets 993.7: Power Sums 1354.1: View Distance versus Altitude 1494.2: Successive Approximations for 2164FiguresNumber and TitlePage1.1: The Circle, Square, and Equilateral Triangle 171.2: Four Ways to Contemplate a Square 181.3: One Square to Two Triangles 191.4: One Possible Path to Discovery 191.5: General Right Triangle 201.6: Four Bi-Shaded Congruent Rectangles 211.7: A Square Donut within a Square 221.8: The Square within the Square is Still There 221.9: A Discovery Comes into View 231.10: Behold! 241.11: Extreme Differences versus… 257

Figures…continuedNumber and TitlePage2.1: Egyptian Knotted Rope, Circa 2000 BCE 272.2: The First Proof by Pythagoras 292.3: An Alternate Visual Proof by Pythagoras 302.4: Annotated Square within a Square 312.5: Algebraic Form of the First Proof 322.6: A Rectangular Dissection Proof 332.7: Twin Triangle Proof 342.8: Euclid’s Windmill without Annotation 362.9 Pondering Squares and Rectangles 372.10: Annotated Windmill 382.11: Windmill Light 402.12: Euclid’s Converse Theorem 432.13: Liu Hui’s Diagram with Template 452.14: Packing Two Squares into One 462.15: The Stomachion Created by Archimedes 482.16: Kurrah Creates the Brides Chair 492.17: Packing the Bride’s Chair into the Big Chair 502.18: Kurrah’s Operation Transformation 512.19: The Devil’s Teeth 522.20: Truth versus Legend 532.21: Bhaskara’s Real Power 542.22: Leonardo da Vinci’s Symmetry Diagram 552.23: Da Vinci’s Proof in Sequence 562.24: Subtle Rotational Symmetry 572.25: Legendre’s Diagram 582.26: Barry Sutton’s Diagram 592.27: Diagram on Henry Perigal’s Tombstone 612.28: Annotated Perigal Diagram 622.29: An Example of Pythagorean Tiling 642.30: Four Arbitrary Placements… 642.31: Exposing Henry’s Quadrilaterals 652.32: President Garfield’s Trapezoid 662.33: Carolyn’s Cauliflower 712.34: Domain D of F 728

Figures…continuedNumber and TitlePage2.35: Carolyn’s Cauliflower for y = 0 752.36: Behavior of G on IntD 762.37: Domain and Locus of Critical Points 772.38: Logically Equivalent Starting Points 792.39: Walking from Inequality to Equality… 802.40: Shearing a Rectangle 832.41: A Four-Step Shearing Proof 843.1: Three Squares, Three Crosses 883.2: Pythagorean Triples 903.3 Inscribed Circle Theorem 963.4 Three Dimensional Pythagorean Theorem 983.5: Three-Dimensional Distance Formula 1003.6: The Three Pythagorean Means 1013.7: The Two Basic Triangle Formulas 1043.8: Schematic of Hero’s Steam Engine 1043.9: Diagram for Heron’s Theorem 1053.10: Diagram for Pappus’ Theorem 1083.11: Pappus Triple Shear-Line Proof 1093.12: Pappus Meets Pythagoras 1103.13: Diagram for Kurrah’s Theorem 1113.14: Diagram for Stewart’s Theorem 1133.15: Trigonometry via Unit Circle 1183.16: Trigonometry via General Right Triangle 1203.17: The Cosine of the Sum 1223.18: An Intricate Trigonometric Decomposition 1253.19: Setup for the Law of Sines and Cosines 1274.1: Triangular Lake and Solution 1374.2: Pure and Perfect 4x4 Magic Square 1414.3: 4x4 Magic Patterns 1424.4: Pythagorean Magic Squares 1434.5: The Schoolhouse Flagpole 1444.6: Off-Limits Windmill 1454.7 Across the Thorns and Nettles 1464.8: Eratosthenes’s Egypt 1479

Figures…continuedNumber and TitlePage4.9: Eratosthenes Measures the Earth 1484.10: View Distance to Earth’s Horizon 1494.11: Measuring the Moon 1504.12: From Moon to Sun 1534.13: From Sun to Alpha Centauri 1554.14: The Golden Ratio 1574.15: Two Golden Triangles 1584.16: Triangular Phi 1604.17: Pythagorean PI 1614.18: Recursive Hypotenuses 1624.19: Pythagorean Spiral 165E.1: Beauty in Order 166E.2: Curry’s Paradox 167A.0: The Tangram 16910

List of Proofs and DevelopmentsSection and TopicPage1.1: Speculative Genesis of Pythagorean Theorem 232.1: Primary Proof by Pythagoras* 292.1: Alternate Proof by Pythagoras* 302.1: Algebraic Form of Primary Proof* 322.1: Algebraic Rectangular Dissection Proof* 332.1: Algebraic Twin-Triangle Proof* 342.2: Euclid’s Windmill Proof* 392.2: Algebraic Windmill Proof* 402.2: Euclid’s Proof of the Pythagorean Converse 422.3: Liu Hui’s Packing Proof* 452.3: Stomachion Attributed to Archimedes 482.4: Kurrah’s Bride’s Chair 492.4: Kurrah’s Transformation Proof* 502.5: Bhaskara’s Minimal Algebraic Proof* 542.6: Leonardo da Vinci’s Skewed-Symmetry Proof * 562.7: Legendre’s Embedded-Similarity Proof* 582.7: Barry Sutton’s Radial-Similarity Proof* 592.8: Henry Perigal’s Quadrilateral-Dissection Proof*622.8: Two Proofs by Pythagorean Tiling* 642.9: President Garfield’s Trapezoid Proof* 662.10: Cauliflower Proof using Calculus* 712.11: Four-Step Shearing Proof* 843.1: Pythagorean Extension to Similar Areas 883.2: Formula Verification for Pythagorean Triples 913.3: Proof of the Inscribed Circle Theorem 963.4: Three-Dimension Pythagorean Theorem 983.4: Formulas for Pythagorean Quartets 993.4: Three-Dimensional Distance Formula 1003.5: Geometric Development of the Three Means 1013.6: Proof of Heron’s Theorem 1063.6: Proof of Pappus’ General Triangle Theorem 1083.6: Proof of Pythagorean Theorem 110Using Pappus’ Theorem*11

List of Proofs and Developments…continuedSection and TopicPage3.6: Proof of Kurrah’s General Triangle Theorem 1113.6: Proof of Pythagorean Theorem 112Using Kurrah’s Theorem*3.6: Proof of Stewart’s General Triangle Theorem 1133.7: Fundamental Unit Circle Trigonometry3.7: Fundamental Right Triangle Trigonometry 1183.7: Development of Trigonometric Addition 122Formulas3.7: Development of the Law of Sines 1273.7: Development of the Law of Cosines 1283.8: Statement Only of Fermat’s Last Theorem 1323.8: Statement of Euler’s Conjecture and Disproof 1334.1: Triangular Lake—Statement and Solutions 1374.2: 4x4 Magic Squares 1414.2: Pythagorean Magic Squares 1434.3: Heights and Distances on Planet Earth 1444.3: Earth’s Radius and Horizon 1474.3: Moon’s Radius and Distance from Earth 1504.3: Sun’s Radius and Distance from Earth 1524.3: Distance from Sun to Alpha Centauri 1554.4: Development of the Golden Ratio Phi 1574.4: Verification of Two Golden Triangles 1584.4: Development of an Iterative Formula for PI 1624.4: Construction of a Pythagorean Spiral 165* These are actual distinct proofs of the PythagoreanTheorem. This book has 20 such proofs in total.12

PrefaceThe Pythagorean Theorem has been with us for over 4000years and has never ceased to yield its bounty to mathematicians,scientists, and engineers. Amateurs love it in that most new proofsare discovered by amateurs. Without the Pythagorean Theorem,none of the following is possible: radio, cell phone, television,internet, flight, pistons, cyclic motion of all sorts, surveying andassociated infrastructure development, and interstellarmeasurement. The Pythagorean Theorem, Crown Jewel of Mathematicschronologically traces the Pythagorean Theorem from aconjectured beginning, Consider the Squares (Chapter 1), through4000 years of Pythagorean proofs, Four Thousand Years of Discovery(Chapter 2), from all major proof categories, 20 proofs in total.Chapter 3, Diamonds of the Same Mind, presents severalmathematical results closely allied to the Pythagorean Theoremalong with some major Pythagorean “spin-offs” such asTrigonometry. Chapter 4, Pearls of Fun and Wonder, is a potpourri ofclassic puzzles, amusements, and applications. An Epilogue, TheCrown and the Jewels, summarizes the importance of thePythagorean Theorem and suggests paths for further exploration.Four appendices service the reader: A] Greek Alphabet, B]Mathematical Symbols, C] Geometric Foundations, and D] References.For the reader who may need a review of elementary geometricconcepts before engaging this book, Appendix C is highlyrecommended. A Topical Index completes the book.A Word on Formats and Use of SymbolsOne of my interests is poetry, having written andstudied poetry for several years now. If you pick up atextbook on poetry and thumb the pages, you will seepoems interspersed between explanations, explanationsthat English professors will call prose. Prose differs frompoetry in that it is a major subcategory of how language isused. Even to the casual eye, prose and poetry each have adistinct look and feel.13

So what does poetry have to do with mathematics?Any mathematics text can be likened to a poetry text. In it,the author is interspersing two languages: a language ofqualification (English in the case of this book) and alanguage of quantification (the universal language ofalgebra). The way these two languages are interspersed isvery similar to that of the poetry text. When we aredescribing, we use English prose interspersed with anillustrative phrase or two of algebra. When it is time to doan extensive derivation or problem-solving activity—usingthe concise algebraic language—then the whole page (or twoor three pages!) may consist of nothing but algebra. Algebrathen becomes the alternate language of choice used tounfold the idea or solution. The Pythagorean Theorem followsthis general pattern, which is illustrated below by adiscussion of the well-known quadratic formula.2Let ax bx c 0 be a quadratic equation writtenin the standard form as shown with a 0 . Thenax2 bx c 0has two solutions (including complex andmultiple) given by the formula below, called the quadraticformula.x b b2 4ac2a .To solve a quadratic equation, using the quadratic formula,one needs to apply the following four steps considered to bea solution process.1. Rewrite the quadratic equation in standard form.2. Identify the two coefficients and constantterm a , b,&c .3. Apply the formula and solve for the two x values.4. Check your two answers in the original equation.To illustrate this four-step process, we will solve thequadratic equation 2x 2 13x 7 .14

1 : 2x2x2****22 13x 7 13x 7 0 : a 2, b 13,c 7****3 ( 13) : x 13 x ( 13)169 5642(2)13 225 1315x 4 41x { ,7}****422 4(2)( 7) : This step is left to the reader.Taking a look at the text between the two happy-facesymbols , we first see the usual mixture of algebra andprose common to math texts. The quadratic formula itself,being a major algebraic result, is presented first as a standaloneresult. If an associated process, such as solving aquadratic equation, is best described by a sequence ofenumerated steps, the steps will be presented in indented,enumerated fashion as shown. Appendix B provides adetailed list of all mathematical symbols used in this bookalong with explanations.Regarding other formats, italicized 9-font text isused throughout the book to convey special cautionarynotes to the reader, items of historical or personal interest,etc. Rather than footnote these items, I have chosen toplace them within the text exactly at the place where theyaugment the overall discussion.15

Lastly, throughout the book, the reader will notice a threesquaredtriangular figure at the bottom of the page. Onesuch figure signifies a section end; two, a chapter end; andthree, the book end.CreditsNo book such as this is an individual effort. Manypeople have inspired it: from concept to completion.Likewise, many people have made it so from drafting topublishing. I shall list just a few and their contributions.Elisha Loomis, I never knew you except throughyour words in The Pythagorean Proposition; but thank youfor propelling me to fashion an every-person’s updatesuitable for a new millennium. To those great Americans ofmy youth—President John F. Kennedy, John Glenn, NeilArmstrong, and the like—thank you all for inspiring anentire generation to think and dream of bigger things thanthemselves.To my two editors, Curtis and Stephanie Sparks,thank you for helping the raw material achieve fullpublication. This has truly been a family affair.To my wife Carolyn, the Heart of it All, what can Isay. You have been my constant and loving partner forsome 40 years now. You gave me the space to complete thisproject and rejoiced with me in its completion. As always,we are a proud team!John C. SparksOctober 2008Xenia, Ohio16

1) Consider the Squares“If it was good enough for old Pythagoras,It is good enough for me.” UnknownHow did the Pythagorean Theorem come to be atheorem? Having not been trained as mathematicalhistorian, I shall leave the answer to that question to thosewho have. What I do offer in Chapter 1 is a speculative,logical sequence of how the Pythagorean Theorem mighthave been originally discovered and then extended to itspresent form. Mind you, the following idealized accountdescribes a discovery process much too smooth to haveactually occurred through time. Human inventiveness inreality always has entailed plenty of dead ends and falsestarts. Nevertheless, in this chapter, I will play the role ofthe proverbial Monday-morning quarterback and execute aperfect play sequence as one modern-day teacher sees it.Figure 1.1: The Circle, Square, and Equilateral TriangleOf all regular, planar geometric figures, the squareranks in the top three for elegant simplicity, the other twobeing the circle and equilateral triangle, Figure 1.1. Allthree figures would be relatively easy to draw by our distantancestors: either freehand or, more precisely, with a stakeand fixed length of rope.17

For this reason, I would think that the square would be oneof the earliest geometrics objects examined.Note: Even in my own early-sixties high-school days, string, chalk,and chalk-studded compasses were used to draw ‘precise’geometric figures on the blackboard. Whether or not this ranks mewith the ancients is a matter for the reader to decide.So, how might an ancient mathematician study asimple square? Four things immediately come to mymodern mind: translate it (move the position in planarspace), rotate it, duplicate it, and partition it into twotriangles by insertion of a diagonal as shown in Figure 1.2.TranslateReplicateMySquareRotatePartitionFigure 1.2: Four Ways to Contemplate a SquareI personally would consider the partitioning of the square tobe the most interesting operation of the four in that I havegenerated two triangles, two new geometric objects, fromone square. The two right-isosceles triangles so generatedare congruent—perfect copies of each other—as shown onthe next page in Figure 1.3 with annotated side lengthss and angle measurements in degrees.18

s090045 0ss45450450090Figure 1.3: One Square to Two TrianglesFor this explorer, the partitioning of the square into perfecttriangular replicates would be a fascination starting pointfor further exploration. Continuing with our speculativejourney, one could imagine the replication of a partitionedsquare with perhaps a little decorative shading as shown inFigure 1.4. Moreover, let us not replicate just once, butfour times.s12345Figure 1.4: One Possible Path to Discovery19

Now, continue to translate and rotate the four replicated,shaded playing piece pieces as if working a jigsaw puzzle.After spending some trial-and-error time—perhaps a fewhours, perhaps several years—we stop to ponder afascinating composite pattern when it finally meanders intoview, Step 5 in Figure 1.4.Note: I have always found it very amusing to see a concise andlogical textbook sequence [e.g. the five steps shown on the previouspage] presented in such a way that a student is left to believe thatthis is how the sequence actually happened in a historical context.Recall that Thomas Edison had four-thousand failures before finallysucceeding with the light bulb. Mathematicians are no less prone todead ends and frustrations!Since the sum of any two acute angles in any one of0the right triangles is again 90 , the lighter-shaded figurebounded by the four darker triangles (resulting from Step 4)is a square with area double that of the original square.Further rearrangement in Step 5 reveals the fundamentalPythagorean sum-of-squares pattern when the threesquares are used to enclose an empty triangular areacongruent to each of the eight original right-isoscelestriangles.Of the two triangle properties for each littletriangle—the fact that each was right or the fact that eachwas isosceles—which was the key for the sum of the twosmaller areas to be equal to the one larger area? Or, wereboth properties needed? To explore this question, we willstart by eliminating one of the properties, isosceles; in orderto see if this magical sum-of-squares pattern still holds.Figure 1.5 is a general right triangle where the threeinterior angles and side lengths are labeled.ACBFigure 1.5: General Right Triangle20

Notice that the right-triangle property implies that the sumof the two acute interior angles equals the right angle asproved below. 180 900 900 0&Thus, for any right triangle, the sum of the two acute angles0equals the remaining right angle or 90 ; eloquently statedin terms of Figure 1.5 as .Continuing our exploration, let’s replicate thegeneral right triangle in Figure 1.5 eight times, dropping allalgebraic annotations. Two triangles will then be fusedtogether in order to form a rectangle, which is shaded viathe same shading scheme in Figure 1.4. Figure 1.6 showsthe result, four bi-shaded rectangles mimicking the four bishadedsquares in Figure 1.4.Figure 1.6: Four Bi-Shaded Congruent RectanglesWith our new playing pieces, we rotate as before, finallyarriving at the pattern shown in Figure 1.7.21

Figure 1.7: A Square Donut within a SquareThat the rotated interior quadrilateral—the ‘square donut’—is indeed a square is easily shown. Each interior cornerangle associated with the interior quadrilateral is part of a0three-angle group that totals 180 . The two acute angles0flanking the interior corner angle sum to 90 since theseare the two different acute angles associated with the righttriangle. Thus, simple subtraction gives the measure of any0one of the four interior corners as 90 . The four sides ofthe quadrilateral are equal in length since they are simplyfour replicates of the hypotenuse of our basic right triangle.Therefore, the interior quadrilateral is indeed a squaregenerated from our basic triangle and its hypotenuse.Suppose we remove the four lightly shaded playingpieces and lay them aside as shown in Figure 1.8.Figure 1.8: The Square within the Square is Still There22

The middle square (minus the donut hole) is still plainlyvisible and nothing has changed with respect to size ororientation. Moreover, in doing so, we have freed up fourplaying pieces, which can be used for further explorations.If we use the four lighter pieces to experiment withdifferent ways of filling the outline generated by the fourdarker pieces, an amazing discover will eventually manifestitself—again, perhaps after a few hours of fiddling andtwiddling or, perhaps after several years—Figure 1.9.Note: To reiterate, Thomas Edison tried 4000 different light-bulbfilaments before discovering the right material for such anapplication.Figure 1.9: A Discovery Comes into ViewThat the ancient discovery is undeniable is plain fromFigure 1.10 on the next page, which includes yet anotherpattern and, for comparison, the original square shown inFigure 1.7 comprised of all eight playing pieces. The 12 thcentury Indian mathematician Bhaskara was alleged tohave simply said, “Behold!” when showing these diagramsto students. Decoding Bhaskara’s terseness, one can createfour different, equivalent-area square patterns using eightcongruent playing pieces. Three of the patterns use half ofthe playing pieces and one uses the full set. Of the threepatterns using half the pieces, the sum of the areas for thetwo smaller squares equals the area of the rotated square inthe middle as shown in the final pattern with the threeoutlined squares.23

Figure 1.10: Behold!Phrasing Bhaskara’s “proclamation” in modern algebraicterms, we would state the following:The Pythagorean TheoremSuppose we have a right triangle with side lengthsand angles labeled as shown below.ACBThen A and2 B2 C224

Our proof in Chapter 1 has been by visual inspection andconsideration of various arrangements of eight triangularplaying pieces. I can imagine our mathematically mindedancestors doing much the same thing some three to fourthousand years ago when this theorem was first discoveredand utilized in a mostly pre-algebraic world.To conclude this chapter, we need to address oneloose end. Suppose we have a non-right triangle. Does thePythagorean Theorem still hold? The answer is aresounding no, but we will hold off proving what is knownas the converse of the Pythagorean Theorem,A2 B2 C2 , until Chapter 2. However, wewill close Chapter 1 by visually exploring two extreme cases2 2 2where non-right angles definitely imply that A B C .Pivot Point to Perfection BelowFigure 1.11: Extreme Differences VersusPythagorean Perfection25

In Figure 1.11, the lightly shaded squares in the upperdiagram form two equal sides for two radically differentisosceles triangles. One isosceles triangle has a large centralobtuse angle and the other isosceles triangle has a smallcentral acute angle. For both triangles, the darker shadedsquare is formed from the remaining side. It is obvious tothe eye that two light areas do not sum to a dark area nomatter which triangle is under consideration. By way ofcontrast, compare the upper diagram to the lower diagramwhere an additional rotation of the lightly shaded squarescreates two central right angles and the associatedPythagorean perfection.Euclid Alone Has Looked on Beauty BareEuclid alone has looked on Beauty bare.Let all who prate of Beauty hold their peace,And lay them prone upon the earth and ceaseTo ponder on themselves, the while they stareAt nothing, intricately drawn nowhereIn shapes of shifting lineage; let geeseGabble and hiss, but heroes seek releaseFrom dusty bondage into luminous air.O blinding hour, O holy, terrible day,When first the shaft into his vision shownOf light anatomized! Euclid aloneHas looked on Beauty bare. Fortunate theyWho, though once and then but far away,Have heard her massive sandal set on stone.Edna St. Vincent Millay26

2) Four Thousand Years of DiscoveryConsider old Pythagoras,A Greek of long ago,And all that he did give to us,Three sides whose squares now showIn houses, fields and highways straight;In buildings standing tall;In mighty planes that leave the gate;And, micro-systems small.Yes, all because he got it rightWhen angles equal ninety—One geek (BC), his plain delight—One world changed aplenty! January 20022.1) Pythagoras and the First ProofPythagoras was not the first in antiquity to knowabout the remarkable theorem that bears his name, but hewas the first to formally prove it using deductive geometryand the first to actively ‘market’ it (using today’s terms)throughout the ancient world. One of the earliest indicatorsshowing knowledge of the relationship between righttriangles and side lengths is a hieroglyphic-style picture,Figure 2.1, of a knotted rope having twelve equally-spacedknots.Figure 2.1: Egyptian Knotted Rope, Circa 2000 BCE27

The rope was shown in a context suggesting its use as aworkman’s tool for creating right angles, done via thefashioning of a 3-4-5 right triangle. Thus, the Egyptianshad a mechanical device for demonstrating the converse ofthe Pythagorean Theorem for the 3-4-5 special case:3 2 2 20 4 5 90 .Not only did the Egyptians know of specificinstances of the Pythagorean Theorem, but also theBabylonians and Chinese some 1000 years beforePythagoras definitively institutionalized the general resultcirca 500 BCE. And to be fair to the Egyptians, Pythagorashimself, who was born on the island of Samos in 572 BCE,traveled to Egypt at the age of 23 and spent 21 years thereas a student before returning to Greece. While in Egypt,Pythagoras studied a number of things under the guidanceof Egyptian priests, including geometry. Table 2.1 brieflysummarizes what is known about the Pythagorean Theorembefore Pythagoras.Date Culture Person Evidence2000BCE1500BCE1100BCE520BCEEgyptianBabylonian&ChaldeanChineseGreekUnknownUnknownTschou-GunPythagorasTable 2.1: Prior to PythagorasWorkman’s rope forfashioning a3-4-5 triangleRules for righttriangles written onclay tablets alongwith geometricdiagramsWritten geometriccharacterizations ofright anglesGeneralized resultand deductivelyproved28

The proof Pythagoras is thought to have actuallyused is shown in Figure 2.2. It is a visual proof in that noalgebraic language is used to support numerically thedeductive argument. In the top diagram, the ancientobserver would note that removing the eight congruent righttriangles, four from each identical master square, brings themagnificent sum-of-squares equality into immediate view.Figure 2.2: The First Proof by Pythagoras29

Figure 2.3 is another original, visual proofattributed to Pythagoras. Modern mathematicians wouldsay that this proof is more ‘elegant’ in that the samedeductive message is conveyed using one less triangle. Eventoday, ‘elegance’ in proof is measured in terms of logicalconciseness coupled with the amount insight provided bythe conciseness. Without any further explanation on mypart, the reader is invited to engage in the mental deductivegymnastics needed to derive the sum-of-squares equalityfrom the diagram below.Figure 2.3: An Alternate Visual Proof by PythagorasNeither of Pythagoras’ two visual proofs requires theuse of an algebraic language as we know it. Algebra in itsmodern form as a precise language of numericalquantification wasn’t fully developed until the Renaissance.The branch of mathematics that utilizes algebra to facilitatethe understanding and development of geometric conceptsis known as analytic geometry. Analytic geometry allows fora deductive elegance unobtainable by the use of visualgeometry alone. Figure 2.4 is the square-within-the-square(as first fashioned by Pythagoras) where the length of eachtriangular side is algebraically annotated just one time.30

cabcFigure 2.4: Annotated Square within a SquareThe proof to be shown is called a dissection proof due to thefact that the larger square has been dissected into fivesmaller pieces. In all dissection proofs, our arbitrary righttriangle, shown on the left, is at least one of the pieces. Oneof the two keys leading to a successful dissection proof isthe writing of the total area in two different algebraic ways:as a singular unit and as the sum of the areas associatedwith the individual pieces. The other key is the need toutilize each critical right triangle dimension—a, b, c—atleast once in writing the two expressions for area. Once thetwo expressions are written, algebraic simplification willlead (hopefully) to the Pythagorean Theorem. Let us startour proof. The first step is to form the two expressions forarea.1 : AAAbigsquarebigsquarebigsquare A c ( a b)littlesquare2 4(122 4 ( Aab)&onetriangle) The second step is to equate these expressions andalgebraically simplify.2 : ( a b)aa22 2ab b b2 cset22 c22 c 4(212ab) 2ab31

Notice how quickly and easily our result is obtained oncealgebraic is used to augment the geometric picture. Simplyput, algebra coupled with geometry is superior to geometryalone in quantifying and tracking the diverse and subtlerelationships between geometric whole and the assortedpieces. Hence, throughout the remainder of the book,analytic geometry will be used to help prove and developresults as much as possible.Since the larger square in Figure 2.4 is dissectedinto five smaller pieces, we will say that this is a DissectionOrder V (DRV) proof. It is a good proof in that all threecritical dimensions—a, b, c—and only these dimensions areused to verify the result. This proof is the proof mostcommonly used when the Pythagorean Theorem is firstintroduced. As we have seen, the origins of this proof can betraced to Pythagoras himself.We can convey the proof in simpler fashion bysimply showing the square-within-the-square diagram(Figure 2.5) and the associated algebraic developmentbelow unencumbered by commentary. Here forward, thiswill be our standard way of presenting smaller and moreobvious proofs and/or developments.abcFigure 2.5: Algebraic Form of the First Proof1:A ( a b)2:(a b) a2set22 c2 2ab b& A c 4(212 c22ab) 4(12ab) 2ab a2 b2 c232

Figure 2.6 is the diagram for a second not-so-obviousdissection proof where a rectangle encloses the basic righttriangle as shown. The three triangles comprising therectangle are similar (left to reader to show), allowing theunknown dimensions x, y, z to be solved via similarityprinciples in terms of a, b, and c. Once we have x, y, and zin hand, the proof proceeds as a normal dissection.zyacbxFigure 2.6: A Rectangular Dissection Proof1x a ab y: x ,b c c b2z a a z a c c2ab:A c ab c 3:A 124 setab b:ab 2 cb2ab abc2 2b a1 2 2c c2 2a b c2abb c c22222a1cbc ab a1c2222ac ac b 2 c22222b b y ,&c cabc 1 2 ab b 2 c22a1c2222a1c 2233

As the reader can immediately discern, this proof, aDRIII, is not visually apparent. Algebra must be used alongwith the diagram in order to quantify the neededrelationships and carry the proof to completion.Note: This is not a good proof for beginning students—say youraverage eighth or ninth grader—for two reasons. One, the algebra issomewhat extensive. Two, derived and not intuitively obviousquantities representing various lengths are utilized to formulate thevarious areas. Thus, the original Pythagorean proof remainssuperior for introductory purposes.Our last proof in this section is a four-step DRVdeveloped by a college student, Michelle Watkins (1997),which also requires similarity principles to carry the proofto completion. Figure 2.7 shows our two fundamental,congruent right triangles where a heavy dashed lineoutlines the second triangle. The lighter dashed linecompletes a master triangle ABC for which we willcompute the area two using different methods. The reader isto verify that each right triangle created by the merger of thecongruent right triangles is similar to the original righttriangle.Step 1 is to compute the length of line segment xusing similarity principles. The two distinct areacalculations in Steps 2 and 3 result from viewing the mastertriangle as either ABC or CBA . Step 4 sets the equalityand completes the proof.CAcabxBFigure 2.7: Twin Triangle Proof34

1x:b2:Area(CBA)3:Area( ABC)Area(ABC)4:c212c aset2 122ba2b x a b{ a2212 b a( c)(c)2 b( a)a aa( a)221212} b22 ba c2122 c2 12{ a22 b }One of the interesting features of this proof is thateven though it is a DRV, the five individual areas were notall needed in order to compute the area associated withtriangle ABC in two different ways. However, some areaswere critical in a construction sense in that they allowed forthe determination of the critical parameter x. Other areastraveled along as excess baggage so to speak. Hence, wecould characterize this proof as elegant but a tad inefficient.However, our Twin Triangle Proof did allow for theintroduction of the construction principle, a principle thatEuclid exploited fully in his great Windmill Proof, thesubject of our next section.35

2.2) Euclid’s Wonderful WindmillEuclid, along with Archimedes and Apollonius, isconsidered one of the three great mathematicians ofantiquity. All three men were Greeks, and Euclid was theearliest, having lived from approximately 330BCE to275BCE. Euclid was the first master mathematics teacherand pedagogist. He wrote down in logical systematic fashionall that was known about plane geometry, solid geometry,and number theory during his time. The result is a treatiseknown as The Elements, a work that consists of 13 booksand 465 propositions. Euclid’s The Elements is one of mostwidely read books of all times. Great minds throughouttwenty-three centuries (e.g. Bertrand Russell in the 20 thcentury) have been initiated into the power of criticalthinking by its wondrous pages.Figure 2.8: Euclid’s Windmill without Annotation36

Euclid’s proof of the Pythagorean Theorem (Book 1,Proposition 47) is commonly known as the Windmill Proofdue to the stylized windmill appearance of the associatedintricate geometric diagram, Figure 2.8.Note: I think of it as more Art Deco.There is some uncertainty whether or not Euclid wasthe actual originator of the Windmill Proof, but that is reallyof secondary importance. The important thing is that Euclidcaptured it in all of elegant step-by-step logical elegance viaThe Elements. The Windmill Proof is best characterized as aconstruction proof as apposed to a dissection proof. InFigure 2.8, the six ‘extra’ lines—five dashed and one solid—are inserted to generate additional key geometric objectswithin the diagram needed to prove the result. Not allgeometric objects generated by the intersecting lines areneeded to actualize the proof. Hence, to characterize theassociated proof as a DRXX (the reader is invited to verifythis last statement) is a bit unfair.How and when the Windmill Proof first came intobeing is a topic for historical speculation. Figure 2.9reflects my personal view on how this might have happened.Figure 2.9: Pondering Squares and Rectangles37

First, three squares were constructed, perhaps bythe old compass and straightedge, from the three sides ofour standard right triangle as shown in Figure 2.9. Theresulting structure was then leveled on the hypotenusesquare in a horizontal position. The stroke of intuitivegenius was the creation of the additional line emanatingfrom the vertex angle and parallel to the vertical sides of thesquare. So, with this in view, what exactly was the beholdersuppose to behold?My own intuition tells me that two complimentaryobservations were made: 1) the area of the lightly-shadedsquare and rectangle are identical and 2) the area of thenon-shaded square and rectangle are identical. Perhapsboth observations started out as nothing more than acurious conjecture. However, subsequent measurements forspecific cases turned conjecture into conviction andinitiated the quest for a general proof. Ancient Greek geniusfinally inserted (period of time unknown) two additionaldashed lines and annotated the resulting diagram as shownin Figure 2.10. Euclid’s proof follows on the next page.FHGEIJKDABCFigure 2.10: Annotated Windmill38

First we establish that the two triangles IJD and GJA arecongruent.1:IJ JG,JD JA,&IJD GJAIJD GJA2:IJD GJAArea(IJD) Area(GJA)2Area(IJD) 2Area(GJA)The next step is to establish that the area of the squareIJGH is double the area of IJD . This is done by carefullyobserving the length of the base and associated altitude foreach. Equivalently, we do a similar procedure for rectangleJABK and GJA .Thus:3:Area(IJGH) 2Area(IJD)4:Area(JABK) 2Area(GJA)5:Area(IJGH) Area(JABK)The equivalency of the two areas associated with the squareGDEF and rectangle BCDK is established in like fashion(necessitating the drawing of two more dashed lines aspreviously shown in Figure 2.8). With this last result, wehave enough information to bring to completion Euclid’smagnificent proof.6:Area(IJGH) Area(GDEF)Area(JABK) Area(BCDK)Area(IJGH) Area(GDEF) Area(ACDJ ) Modern analytic geometry greatly facilitates Euclid’scentral argument. Figure 2.11 is a much-simplifiedwindmill with only key dimensional lengths annotated..39

axycbFigure 2.11: Windmill LightThe analytic-geometry proof below rests on thecentral fact that the two right triangles created by theinsertion of the perpendicular bisector are both similar tothe original right triangle. First, we establish the equality ofthe two areas associated with the lightly shaded square andrectangle via the following logic sequence:1:2xa:A3:A4:A2a a x c cshadedsquareshadedrectshadedsquare aac2A2c ashadedrect2.Likewise, for the non-shaded square and rectangle:40

1 2y b b: y b c c2:A3:A4:Aunshadedsquareunshadedrectunshadedsquare b2bcA2c b2unshadedrectPutting the two pieces together (quite literally), we have:12 : A bigsquare c 2 2 2 2 2c a b a b cThe reader probably has discerned by now thatsimilarity arguments play a key role in many proofs of thePythagorean Theorem. This is indeed true. In fact, proof bysimilarity can be thought of a major subcategory just likeproof by dissection or proof by construction. Proof byvisualization is also a major subcategory requiring crystalclear,additive dissections in order to make the PythagoreanTheorem visually obvious without the help of analyticgeometry. Similarity proofs were first exploited in wholesalefashion by Legendre, a Frenchman that had the full powerof analytic geometry at his disposal. In Section 2.7, we willfurther reduce Euclid’s Windmill to its primal bare-bonesform via similarity as first exploited by Legendre.Note: More complicated proofs of the Pythagorean Theorem usuallyare a hybrid of several approaches. The proof just given can bethought of as a combination of construction, dissection, andsimilarity. Since similarity was the driving element in forming theargument, I would primarily characterize it as a similarity proof.Others may characteristic it as a construction proof since noargument is possible without the insertion of the perpendicularbisector. Nonetheless, creation of a perpendicular bisection creates adissection essential to the final addition of squares and rectangles!Bottom line: all things act together in concert.2.41

We close this section with a complete restatement of thePythagorean Theorem as found in Chapter 2, but now withthe inclusion of the converse relationshipA2 B2 C2 900. Euclid’s subtle proof of thePythagorean Converse follows (Book 1 of The Elements,Proposition 48).The Pythagorean Theorem andPythagorean ConverseSuppose we have a triangle with side lengthsand angles labeled as shown below.Then: CA B A2 B2 C2Figure 2.12 on the next page shows Euclid’s originalconstruction used to prove the Pythagorean Converse. Theshaded triangle conforms to the hypothesisA2 B2 C2whereby design. From the intentional design,0one is to show or deduce that 90 in orderto prove the converse.Note: the ancients and even some of my former public-schoolteachers would have said ‘by construction’ instead of ‘by design.’However, the year is 2008, not 1958, and the word design seems tobe a superior conveyor of the intended meaning.42

XB'A090 CBFigure 2.12: Euclid’s Converse DiagramEuclid’s first step was to construct a line segment oflength B ' where B' B . Then, this line segment was joinedas shown to the shaded triangle in such a fashion that thecorner angle that mirrors was indeed a right angle ofmeasure—again by design! Euclid then added a second lineof unknown length X in order to complete a companiontriangle with common vertical side as shown in Figure2.12. Euclid finally used a formal verbally-descriptive logicstream quite similar to the annotated algebraic logic streambelow in order to complete his proof.Algebraic Logic1: A2 B2 C 22:B B'3: B'A 9042 2 2: A ( B') X5: XXX222 A CX C A222 B0 ( B')22Verbal Annotation1] By hypothesis2] By design3] By design4] Pythagorean Theorem5] Properties of algebraicequality90043

6:ABC AB'X6] The three correspondingsides are equal in length (SSS)70: 907] The triangles ABC8: 180 90 9000 18000 and AB' X are congruent8] Properties of algebraicequalityWe close this section by simply admiring the simple andprofound algebraic symmetry of the Pythagorean Theoremand its converse as ‘chiseled’ belowCB AA2 B2 C244

2.3) Liu Hui Packs the SquaresLiu Hui was a Chinese philosopher andmathematician that lived in the third century ACE. By thattime, the great mathematical ideas of the Greeks wouldhave traveled the Silk Road to China and visa-versa, withthe cross-fertilization of two magnificent cultures enhancingthe further global development of mathematics. As justdescribed, two pieces of evidence strongly suggest thatindeed this was the case.Figure 2.13 is Liu Hui’s exquisite diagramassociated with his visual proof of the PythagoreanTheorem.Figure 2.13: Liu Hui’s Diagram with Template45

In it, one clearly sees the Greek influence of Pythagoras andEuclid. However, one also sees much, much more: a moreintricate and clever visual demonstration of the‘Pythagorean Proposition’ than those previouslyaccomplished.Note: Elisha Loomis, a fellow Ohioan, whom we shall meet inSection 2.10, first used the expression ‘Pythagorean Proposition’over a century ago.Is Liu Hui’s diagram best characterized as adissection (a humongous DRXIIII not counting the blackright triangle) or a construction?Figure 2.14: Packing Two Squares into OneWe will say neither although the diagram haselements of both. Liu Hui’s proof is best characterized as apacking proof in that the two smaller squares have beendissected in such a fashion as to allow them to packthemselves into the larger square, Figure 2.14. Some yearsago when our youngest son was still living at home, webought a Game Boy and gave it to him as a Christmaspresent. In time, I took a liking to it due to the nifty puzzlegames.46

One of my favorites was Boxel, a game where the playerhad to pack boxes into a variety of convoluted warehouseconfigurations. In a sense, I believe this is precisely whatLiu Hui did: he perceived the Pythagorean Proposition as apacking problem and succeeded to solve the problem by themasterful dismemberment and reassembly as shown above.In the spirit of Liu Hui, actual step-by-step confirmation of the‘packing of the pieces’ is left to the reader as a challengingvisual exercise.Note: One could say that Euclid succeeded in packing two squaresinto two rectangles, the sum of which equaled the square formed onthe hypotenuse.So what might have been the origin of Liu Hui’spacking idea? Why did Liu Hui use such odd-shaped pieces,especially the two obtuse, scalene triangles? Finally, whydid Liu Hui dissect the three squares into exactly fourteenpieces as opposed to twenty? Archimedes (287BCE-212BCE), a Greek and one of the three greatestmathematicians of all time—Isaac Newton and Karl Gaussbeing the other two—may provide some possible answers.Archimedes is commonly credited (rightly orwrongly) with a puzzle known by two names, theArchimedes’ Square or the Stomachion, Figure 2.15 on thenext page. In the Stomachion, a 12 by 12 square grid isexpertly dissected into 14 polygonal playing pieces whereeach piece has an integral area. Each of the fourteen piecesis labeled with two numbers. The first is the number of thepiece and the second is the associated area. Two views ofthe Stomachion are provided in Figure 2.15, an ‘artist’sconcept’ followed by an ‘engineering drawing’. I would liketo think that the Stomachion somehow played a key role inLiu Hui’s development of his magnificent packing solutionto the Pythagorean Proposition. Archimedes’ puzzle couldhave traveled the Silk Road to China and eventually foundits way into the hands of another ancient and great out-ofthe-boxthinker!47

1, 129, 242, 123, 127, 68, 1210, 3111, 95, 36, 2113, 612, 64, 614, 121Figure 2.15: The Stomachion Created by Archimedes48

2.4) Kurrah Transforms the Bride’s ChairOur youngest son loved Transformers as a child,and our oldest son was somewhat fond of them too. Forthose of you who may not remember, a transformer is amechanical toy that can take on a variety of shapes—e.g.from truck to robot to plane to boat—depending how youtwist and turn the various appendages. The idea oftransforming shapes into shapes is not new, even thoughthe 1970s brought renewed interest in the form of highlymarketable toys for the children of Baby Boomers. Eventoday, a new league of Generation X parents are digging intheir pockets and shelling out some hefty prices for thoseirresistible Transformers.Thabit ibn Kurrah (836-901), a Turkish-bornmathematician and astronomer, lived in Baghdad duringIslam’s Era of Enlightenment paralleling Europe’s DarkAges. Kurrah (also Qurra or Qorra) developed a clever andoriginal proof of the Pythagorean Theorem along with a nonrighttriangle extension of the same (Section 3.6) Kurrah’sproof has been traditionally classified as a dissection proof.Then again, Kurrah’s proof can be equally classified as atransformer proof. Let’s have a look.The Bride’s ChairFigure 2.16: Kurrah Creates the Bride’s Chair49

Figure 2.16 shows Kurrah’s creation of the Bride’sChair. The process is rather simple, but shows Kurrah’sintimate familiarity with our fundamental Pythagoreangeometric structure on the left. Four pieces comprise thebasic structure and these are pulled apart and rearrangedas depicted. The key rearrangement is the one on the topright that reassembles the two smaller squares into a newconfiguration known as the bride’s chair. Where the name‘Bride’s Chair’ originated is a matter for speculation;personally, I think the chair-like structure looks more like aLazy Boy.Now what? Kurrah had a packing problem—twolittle squares to be packed into one big square—which hecleverly solved by the following dissection and subsequenttransformation. Figure 2.17 pictorially captures Kurrah’sdilemma, and his key dissection that allowed thetransformation to proceed.?The Bride’s Chair!Figure 2.17: Packing the Bride’s Chair intoThe Big SquareWhat Kurrah did was to replicate the shaded triangle anduse it to frame two cutouts on the Bride’s Chair as shown.Figure 2.18 is ‘Operation Transformation’ showingKurrah’s rotational sequence that leads to a successfulpacking of the large square.50

Note: as is the occasional custom in this volume, the reader is askedto supply all dimensional details knowing that the diagram isdimensionally correct. I am convinced that Kurrah himself wouldhave demanded the same.PP!PP!P is a fixedpivot pointPP!Figure 2.18: Kurrah’s Operation TransformationAs Figure 2.18 clearly illustrates, Kurrah took a cleverlydissected Bride’s Chair and masterfully packed it into thebig square though a sequence of rotations akin to thoseemployed by the toy Transformers of today—ademonstration of pure genius!51

The Bride’s Chair and Kurrah’s subsequentdissection has long been the source for a little puzzle thathas found its way into American stores for at least fortyyears. I personally dub this puzzle ‘The Devil’s Teeth’,Figure 2.19. As one can see, it nothing more than theBride’s Chair cut into four pieces, two of which are identicalright triangles. The two remaining pieces are arbitrarily cutfrom the residual of the Bride’s Chair. Figure 2.19 depictstwo distinct versions of ‘The Devil’s Teeth’.Figure 2.19: ‘The Devil’s Teeth’The name ‘Devil’s Teeth’ is obvious: the puzzle is a devilishone to reassemble. If one adds a little mysticism about thesignificance of the number four, you probably got a winneron your hands. In closing, I can imagine Paul Harvey doing aradio spot focusing on Kurrah, the Bride’s Chair, and thethousand-year-old Transformers proof. After thecommercial break, he describes ‘The Devil’s Teeth’ and thesuccessful marketer who started this business out of agarage. “And that is the rest of the story. Good day.”Note: I personally remember this puzzle from the early 1960s.52

2.5) Bhaskara Unleashes the Power of AlgebraWe first met Bhaskara in Chapter 2. He was the 12 thcentury (circa 1115 to 1185) Indian mathematician whodrew the top diagram shown in Figure 2.20 and simplysaid, “Behold!”, completing his proof of the PythagoreanTheorem. However, legend has a way of altering details andfish stories often times get bigger. Today, what is commonlyascribed to Bhaskara’s “Behold!” is nothing more than thenon-annotated square donut in the lower diagram. I, forone, have a very hard time beholding exactly what I amsuppose to behold when viewing the non-annotated squaredonut. Appealing to Paul Harvey’s famous radio format asecond time, perhaps there is more to this story. There is.Bhaskara had at his disposal a well-developed algebraiclanguage, a language that allowed him to capture preciselyvia analytic geometry those truths that descriptive geometryalone could not easily convey.Figure 2.20: Truth Versus Legend53

What Bhaskara most likely did as an accomplishedalgebraist was to annotate the lower figure as shown againin Figure 2.21. The former proof easily follows in a fewsteps using analytic geometry. Finally, we are ready for thefamous “Behold! as Bhaskara’s magnificent DRVPythagorean proof unfolds before our eyes.bac1 : AAA2 : ccc22Figure 2.21: Bhaskara’s Real Powerbigsquarebigsquare a abigsquareset2 A ( a b)(a b)22 2ab b b2 c22 a&littlesquare2 4( 4(22 4(A12 2ab bab)212ab) conetriangle2) Bhaskara’s proof is minimal in that the large square hasthe smallest possible linear dimension, namely c. It alsoutilizes the three fundamental dimensions—a, b, & c—asthey naturally occur with no scaling or proportioning. Thetricky part is size of the donut hole, which Bhaskara’s useof analytic geometry easily surmounts. Thus, only one wordremains to describe this historic first—behold!54

2.6) Leonardo da Vinci’s Magnificent SymmetryLeonardo da Vinci (1452-1519) was born inAnchiano, Italy. In his 67 years, Leonardo became anaccomplished painter, architect, designer, engineer, andmathematician. If alive today, the whole world wouldrecognize Leonardo as ‘world class’ in all theaforementioned fields. It would be as if Stephen Hawkingand Stephen Spielberg were both joined into one person.For this reason, Leonardo da Vinci is properly characterizedas the first and greatest Renaissance man. The world hasnot seen his broad-ranging intellectual equivalent since!Thus, it should come as no surprise that Leonardo da Vinci,the eclectic master of many disciplines, would havethoroughly studied and concocted an independent proof ofthe Pythagorean Theorem.Figure 2.22 is the diagram that Leonardo used todemonstrate his proof of the Pythagorean Proposition. Theadded dotted lines are used to show that the right angle ofthe fundamental right triangle is bisected by the solid linejoining the two opposite corners of the large dotted squareenclosing the lower half of the diagram. Alternately, the twodotted circles can also be used to show the same (readerexercise).Figure 2.22: Leonardo da Vinci’s Symmetry Diagram55

Figure 2.23 is the six-step sequence that visuallydemonstrates Leonardo’s proof. The critical step is Step 5where the two figures are acknowledged by the observer tobe equivalent in area. Step 6 immediately follows.123456Figure 2.23: Da Vinci’s Proof in Sequence56

In Figure 2.24, we enlarge Step 5 and annotate the criticalinternal equalities. Figure 2.24 also depicts the subtlerotational symmetry between the two figures by labeling thepivot point P for an out-of-plane rotation where the lower0half of the top diagram is rotated 180 in order to matchthe bottom diagram. The reader is to supply the supportingrationale. While doing so, take time to reflect on the subtleand brilliant genius of the Renaissance master—Leonardoda Vinci!b450cbcbPcaaa045b045bcbcccacaa045acacbcbFigure 2.24: Subtle Rotational Symmetry57

2.7) Legendre Exploits Embedded SimilarityAdrian Marie Legendre was a well-known Frenchmathematician born at Toulouse in 1752. He died at Parisin 1833. Along with Lagrange and Laplace, Legendre can beconsidered on of the three fathers of modern analyticgeometry, a geometry that incorporates all the inherentpower of both algebra and calculus. With much of his life’swork devoted to the new analytic geometry, it should comeas no surprise that Legendre should be credited with apowerful, simple and thoroughly modern—for the time—new proof of the Pythagorean Proposition. Legendre’s proofstarts with the Windmill Light (Figure 2.11). Legendre thenpared it down to the diagram shown in Figure 2.25.axycbFigure 2.25: Legendre’s DiagramHe then demonstrated that the two right triangles formed bydropping a perpendicular from the vertex angle to thehypotenuse are both similar to the master triangle. Armedwith this knowledge, a little algebra finished the job.12x a a: x a c c22y b b: y b c c3:x y c ac22bc c a2 b2 cNotice that this is the first proof in our historical sequencelacking an obvious visual component.258

But, this is precisely the nature of algebra and analyticgeometry where abstract ideas are more precisely (andabstractly) conveyed than by descriptive (visual) geometryalone. The downside is that visual intuition plays a minimalrole as similarity arguments produce the result via a fewalgebraic pen strokes. Thus, this is not a suitable beginner’sproof.Similarity proofs have been presented throughoutthis chapter, but Legendre’s is historically the absoluteminimum in terms of both geometric augmentation (thedrawing of additional construction lines, etc.) and algebraicterseness. Thus, it is included as a major milestone in oursurvey of Pythagorean proofs. To summarize, Legendre’sproof can be characterized as an embedded similarity proofwhere two smaller triangles are created by the dropping ofjust one perpendicular from the vertex of the mastertriangle. All three triangles—master and the two created—are mutually similar. Algebra and similarity principlescomplete the argument in a masterful and modern way.Note: as a dissection proof, Legendre’s proof could be characterizedas a DRII, but the visual dissection is useless without the powerfulhelp of algebra, essential to the completion of the argument.Not all similarity proofs rely on complicated ratiossuch as2x a / c to evaluate constructed lineardimensions in terms of the three primary quantities a, b, &c. Figure 2.26 is the diagram recently used (2002) by J.Barry Sutton to prove the Pythagorean Proposition usingsimilarity principles with minimally altered primaryquantities.EAac-bBcbbCbDCircle of radius bFigure 2.26 Barry Sutton’s Diagram59

We end Section 2.7 by presenting Barry’s proof instep-by-step fashion so that the reader will get a sense ofwhat formal geometric logic streams look like, as they arefound in modern geometry textbooks at the high school orcollege level.1: Construct right triangle AEC with sides a, b, c.2:Construct circle Cb centered at C with radius b.3: Construct triangle4:BED right BED with hypotenuse 2b.By inscribed triangle theorem sincethe hypotenuse for BED equals andexactly overlays the diameter for Cb5:AEB CED The same common angle BEC issubtracted from the right angles AEC and BED6:CED CDE The triangle CED7:AEB CDE Transitivity of equality8:AEB AED The angle DAEtriangles and AEB CDE is isosceles. is common to both. Hencethe third angle is equal and similarityis assured by AAA.With the critical geometric similarity firmly established bytraditional logic, Barry finishes his proof with an algebraiccoup-de-grace that is typical of the modern approach!9AE AB a c b: AD AE c b a2a ( c b)(c b)a2 c2 b2 a2 b2 c2Equality of similar ratios60

2.8) Henry Perigal’s TombstoneHenry Perigal was an amateur mathematician andastronomer who spent most of his long life (1801-1898)near London, England. Perigal was an accountant by trade,but stargazing and mathematics was his passion. He was aFellow of the Royal Astronomical Society and treasurer ofthe Royal Meteorological Society. Found of geometricdissections, Perigal developed a novel proof of thePythagorean Theorem in 1830 based on a rather intricatedissection, one not as transparent to the casual observerwhen compared to some of the proofs from antiquity. Henrymust have considered his proof of the Pythagorean Theoremto be the crowning achievement of his life, for the diagramis chiseled on his tombstone, Figure 2.27. Notice the cleveruse of key letters found in his name: H, P, R, G, and L.PRHCGLALGBHRPAB 2 =AC 2 +CB 2DISCOVERED BY H.P.1830Figure 2.27: Diagram on Henry Perigal’s Tombstone61

In Figure 2.28, we update the annotations used by Henryand provide some key geometric information on his overallconstruction.Master righttriangle a, b, cECFBb 2ADbacGThis is the center ofthe ‘a’ square. Thetwo cross lines runparallel to thecorresponding sidesof the ‘c’ square.b 2IAll eightconstructedquadrilaterals arecongruentHEach line segment isconstructed from themidpoint of a sideand runs parallel tothe correspondingside of the ‘b’ square.Figure 2.28: Annotated Perigal DiagramWe are going to leave the proof to the reader as a challenge.Central to the Perigal argument is the fact that all eight ofthe constructed quadrilaterals are congruent. Thisimmediately leads to fact that the middle square embeddedin the ‘c’ square is identical to the ‘a’ square from which2 2 2a b c can be established.Perigal’s proof has since been cited as one of themost ingenious examples of a proof associated with aphenomenon that modern mathematicians call aPythagorean Tiling.62

In the century following Perigal, both Pythagorean Tilingand tiling phenomena in general were extensively studiedby mathematicians resulting in two fascinating discoveries:1] Pythagorean Tiling guaranteed that the existence ofcountless dissection proofs of the Pythagorean Theorem.2] Many previous dissection proofs were in actuality simplevariants of each, inescapably linked by Pythagorean Tiling.Gone forever was the keeping count of the number of proofsof the Pythagorean Theorem! For classical dissections, thecontinuing quest for new proofs became akin to writing thenumbers from 1 ,234, 567 to 1 ,334, 567 . People started toask, what is the point other than garnering a potential entryin the Guinness Book of World Records? As we continueour Pythagorean journey, keep in mind Henry Perigal, for itwas he (albeit unknowingly) that opened the door to thismore general way of thinking.Note: Elisha Loomis whom we shall meet in Section 2.10, publisheda book in 1927 entitled The Pythagorean Proposition, in which hedetails over 350 original proofs of the Pythagorean Theorem.We are now going to examine Perigal’s novel proofand quadrilateral filling using the modern methodsassociated with Pythagorean Tiling, an example of which isshown in Figure 2.29 on the next page. From Figure 2.29,we see that three items comprise a Pythagorean Tilingwhere each item is generated, either directly or indirectly,from the master right triangle.1) The Bride’s Chair, which serves as a basic tessellationunit when repeatedly drawn.2) The master right triangle itself, which serves as an‘anchor-point’ somewhere within the tessellation pattern.3) A square cutting grid, aligned as shown with thetriangular anchor point. The length of each line segmentwithin the grid equals the length of the hypotenuse for themaster triangle. Therefore, the area of each square holeequals the area of the square formed on the hypotenuse.63

Figure 2.29: An Example of Pythagorean TilingAs Figure 2.29 illustrates, the square cutting gridimmediately visualizes the cuts needed in order to dissectand pack the two smaller squares into the hypotenusesquare, given a particular placement of the black triangle.Figure 2.30 shows four different, arbitrary placements ofthe anchor point that ultimately lead to four dissectionsand four proofs once the cutting grid is properly placed.Bottom line: a different placement means a different proof!Figure 2.30: Four Arbitrary Placements of theAnchor Point64

Returning to Henry Perigal, Figure 2.31 shows theanchor placement and associated Pythagorean Tilingneeded in order to verify his 1830 dissection. Notice howthe viability of Henry Perigal’s proof and novel quadrilateralis rendered immediately apparent by the grid placement. Aswe say in 2008, slick!Figure 2.31: Exposing Henry’s QuadrilateralsWith Pythagorean Tiling, we can have a thousand differentplacements leading to a thousand different proofs. Shouldwe try for a million? Not a problem! Even old Pythagorasand Euclid might have been impressed.65

2.9) President Garfield’s Ingenious TrapezoidThe Ohioan James A. Garfield (1831-1881) was the20 th president of the United States. Tragically, Garfield’sfirst term in office was cut short by an assassin’s bullet:inaugurated on 4 March 1881, shot on 2 July 1881, anddied of complications on 19 Sept 1881. Garfield came frommodest Midwestern roots. However, per hard work he wasable to save enough extra money in order to attendWilliam’s College in Massachusetts. He graduated withhonors in 1856 with a degree in classical studies. After ameteoric stint as a classics professor and (within two years)President of Hiram College in Ohio, Garfield was elected tothe Ohio Senate in 1859 as a Republican. He fought in theearly years of the Civil War and in 1862 obtained the rankof Brigadier General at age 31 (achieving a final rank ofMajor General in 1864). However, Lincoln had other plansfor the bright young Garfield and urged him to run for theU.S. Congress. Garfield did just that and served from 1862to 1880 as a Republican Congressman from Ohio,eventually rising to leading House Republican.While serving in the U.S. Congress, Garfieldfabricated one of the most amazing and simplistic proofs ofthe Pythagorean Theorem ever devised—a dissection proofthat looks back to the original diagram attributed toPythagoras himself yet reduces the number of playingpieces from five to three.cbaFigure 2.32: President Garfield’s Trapezoid66

Figure 2.32 is President Garfield’s Trapezoid diagram inupright position with its origin clearly linked to Figure 2.3.Recall that the area of a trapezoid, in particular the area ofthe trapezoid in Figure 2.32, is given by the formula:A Trap ( a b a b )12.Armed with this information, Garfield completes his proofwith a minimum of algebraic pen strokes as follows.121212121:A2:A3:aa1 ( a b) a b1 ( a b) a baa22222TrapTrap 2ab b 2ab b ab 12b2212ab 12b12222c ab 212c2set121212 aab ab ab 21212abc12122 bcc21222c2 c2121212ab ab ab Note: President Garfield actually published his proof in the 1876edition of the Journal of Education, Volume 3, Issue 161, where thetrapezoid is shown lying on its right side.It does not get any simpler than this! Garfield’s proof is amagnificent DRIII where all three fundamental quantities a,b, c are used in their natural and fundamental sense. Anextraordinary thing is that the proof was not discoveredsooner considering the ancient origins of Garfield’strapezoid. Isaac Newton, the co-inventor of calculus, oncesaid. “If I have seen further, it has been by standing on theshoulders of giants.” I am sure President Garfield, a giant inhis own right, would concur. Lastly, speaking of agreement,Garfield did have this to say about his extraordinary andsimple proof of the Pythagorean Proposition, “This is onething upon which Republicans and Democrats should bothagree.”.67

2.10] Ohio and the Elusive Calculus ProofElisha Loomis (1852-1940), was a Professor ofMathematics, active Mason, and contemporary of PresidentGarfield. Loomis taught at a number of Ohio colleges andhigh schools, finally retiring as mathematics departmenthead for Cleveland West High School in 1923. In 1927,Loomis published a still-actively-cited book entitled ThePythagorean Proposition, a compendium of over 250 proofsof the Pythagorean Theorem—increased to 365 proofs inlater editions. The Pythagorean Proposition was reissued in1940 and finally reprinted by the National Council ofTeachers of Mathematics in 1968, 2 nd printing 1972, as partof its “Classics in Mathematics Education” Series.Per the Pythagorean Proposition, Loomis is creditedwith the following statement; there can be no proof of thePythagorean Theorem using either the methods oftrigonometry or calculus. Even today, this statementremains largely unchallenged as it is still found with sourcecitation on at least two academic-style websites 1 . Forexample, Jim Loy states on his website, “The book ThePythagorean Proposition, by Elisha Scott Loomis, is a fairlyamazing book. It contains 256 proofs of the PythagoreanTheorem. It shows that you can devise an infinite number ofalgebraic proofs, like the first proof above. It shows that youcan devise an infinite number of geometric proofs, likeEuclid's proof. And it shows that there can be no proofusing trigonometry, analytic geometry, or calculus. Thebook is out of print, by the way.”That the Pythagorean Theorem is not provable usingthe methods of trigonometry is obvious since trigonometricrelationships have their origin in a presupposedPythagorean right-triangle condition. Hence, any proof bytrigonometry would be a circular proof and logically invalid.However, calculus is a different matter.1 See Math Forum@ Drexel,http://mathforum.org/library/drmath/view/6259.html ;Jim Loy website,http://www.jimloy.com/geometry/pythag.htm .68

Even though the Cartesian coordinate finds its way intomany calculus problems, this backdrop is not necessary inorder for calculus to function since the primary purpose ofa Cartesian coordinate system is to enhance ourvisualization capability with respect to functional and otheralgebraic relationships. In the same regard, calculus mostdefinitely does not require a metric of distance—as definedby the Distance Formula, another Pythagorean derivate—inorder to function. There are many ways for one to metricizeEuclidean n-space that will lead to the establishment ofrigorous limit and continuity theorems. Table 2.2 lists thePythagorean metric and two alternatives. Reference 19presents a complete and rigorous development of thedifferential calculus for one and two independent variablesusing the rectangular metric depicted in Table 2.2.METRIC SET DEFINITION SHAPEPythagoreanTaxi CabRectangular2( x x0 ) ( y y0)|0y0x x | | y | |02 x x | and y y | |0CircleDiamondSquareTable 2.2 Three Euclidean MetricsLastly, the derivative concept—albeit enhanced viathe geometric concept of slope introduced with a touch ofmetrics—is actually a much broader notion thaninstantaneous “rise over run”. So what mathematicalprinciple may have prompted Elisha Loomis, our early 20 thcentury Ohioan, to discount the methods of calculus as aviable means for proving the Pythagorean Theorem? Onlythat calculus requires geometry as a substrate. The implicitand untrue assumption is that all reality-based geometry isPythagorean. For a realty-based geometric counterexample,the reader is encouraged to examine Taxicab Geometry: anAdventure in Non-Euclidean Geometry by Eugene Krauss,(Reference 20).69

Whatever the original intent or implication, the PythagoreanProposition has most definitely discouraged the quest forcalculus-based proofs of the Pythagorean Theorem, for theyare rarely found or even mentioned on the worldwide web.This perplexing and fundamental void in elementarymathematics quickly became a personal challenge to searchfor a new calculus-based proof of the Pythagorean Theorem.Calculus excels in its power to analyze changing processesincorporating one or more independent variables. Thus, onewould think that there ought to be something of value inIsaac Newton and Gottfried Leibniz’s brainchild—hailed bymany as the greatest achievement of Western science andcertainly equal to the Pythagorean brainchild—that wouldallow for an independent metrics-free investigation of thePythagorean Proposition.Note: I personally remember a copy of the PythagoreanProposition—no doubt, the 1940 edition—sitting on my father’sbookshelf while yet a high-school student, Class of 1965.Initial thoughts/questions were twofold. Couldcalculus be used to analyze a general triangle as itdynamically changed into a right triangle? Furthermore,could calculus be used to analyze the relationship amongst2 2 2the squares of the three sides A , B , C throughout theprocess and establish the sweet spot of equality2 2 2A B C —the Pythagorean Theorem? Being a lifelongOhioan from the Greater Dayton area personallyhistoricized this quest in that I was well aware of thesignificant contributions Ohioans have made to technicalprogress in a variety of fields. By the end of 2004, a viableapproach seemed to be in hand, as the inherent power ofcalculus was unleashed on several ancient geometricstructures dating back to the time of Pythagoras himself.Figure 2.33, Carolyn’s Cauliflower (so named in honor ofmy wife who suggested that the geometric structure lookedlike a head of cauliflower) is the geometric anchor point fora calculus-based proof of the Pythagorean Theorem.70

B 2← γ →α y βx C-xC 2Figure 2.33: Carolyn’s CauliflowerThe goal is to use the optimization techniques ofmultivariable differential calculus to show that the three2 22squares A , B , and C constructed on the three sides ofthe general triangle shown in Figure 2.33, with angles are , , and , satisfy the Pythagorean condition if and only 180 if. Sincefor any triangle, therightmost equality is equivalent to thecondition( , , )10 900, which in turn implies that triangleis a right triangle. : To start the proof , let 0C be the fixed length of anarbitrary line segment placed in a horizontal position. Let xbe an arbitrary point on the line segment which cuts theline segment into two sub-lengths: x and C x . Let0 y C be an arbitrary length of a perpendicular linesegment erected at the point x . Since x and y are botharbitrary, they are both independent variables in the classicsense.71

In addition, y serves as the altitude for the arbitrary triangle( , , ) defined by the construction shown in Figure2.33. The sum of the two square areas2A and2B inFigure 2.33 can be determined in terms of x and y asfollows:The terms2 222A B ( x y) ([ C x] y) 2Cy .2( x y) and([ C y2 x] ) are the areas of theleft and right outer enclosing squares, and the term2 Cy isthe combined area of the eight shaded triangles expressedas an equivalent rectangle. Define F ( x,y)as follows:F2 2 2( x,y) { A B C}2Then, substituting the expression for2A B2, we haveF(x,y) {( x y)F(x,y) {2xF(x,y) 4{ x[C x] y22 2y ([ C x] y)2 2Cx}2}222 2Cy C2}2We now restrict the function F to the compact, squaredomain D {( x,y)| 0 x C,0 y C}shown in Figure2.34 where the symbols BndD and IntD denote theboundary of D and interior of D respectively.(0, C) (C, C)BndDIntD(0, 0)(C, 0)Figure 2.34: Domain D of F72

Being polynomial in form, the function F is bothcontinuous and differentiable on D . Continuity impliesthat F achieves both an absolute maximum and absoluteminimum on D , which occur either on BndD or IntD .Additionally, F ( x,y) 0 for all points ( x,y)in D due tothe presence of the outermost square inF( x,y)y 4{ x[C x]2 } 2 .This implies in turn that F ( xmin, ymin) 0 for point(s)( xmin, ymin) corresponding to absolute minimum(s) forF on D . Equality to zero will be achieved if and only if2minx [ C x ] y 0 .minminReturning to the definition for F , one can immediately seethat the following four expressions are mutually equivalentF(xxAAmin22min, y22min[ C x B Bmin C) 0 ] y2 C22min 0 0 2 : We now employ the optimization methods ofmultivariable differential calculus to search for those points( xmin, ymin) where F ( xmin, ymin) 0 (if such points exist)and study the implications. First we examine F( x,y)forpoints ( x,y)restricted to the four line segments comprisingBndD .1.2F( x,0) 4{ x[C x]}. This implies F ( x,0) 0 onlyx or x C on the lower segment of BndD .when 0Both of these x values lead to degenerate cases.73

2 22. F(x,C) 4{ x[C x] C } 0 for all points on theupper segment of BndD since the smallest value2that | x[C x] C | achieves is 3C 2 / 4 , asdetermined via the techniques of single-variabledifferential calculus.3. F (0, y) F(C,y) 4y4 0 for all points other thany 0 (a degenerate case) on the two verticalsegments of BndD .To examine F on IntD , first take the partial derivatives ofF with respect to x and y . This gives after simplification:F(x,y) / x 8[ C 2x]{x[C x]F(x,y) / y 16y{x[C x]2y}2yNext, set the two partial derivatives equal to zF ( x,y) / x F(x,y) / y 0 .Solving for the associated critical points x , y ) yields}(cp cpone specific critical point and an entire locus of criticalpoints as follows:(C 2 ,0) , a specific critical point2cpx [ C x ] y 0 ,cpcpan entire locus of critical points.The specific critical point (C 2 ,0)is the midpoint of the lowersegment for BndD . We have that4F(C ,0) C / 4 0 .2 74

Thus, the critical point ( C 2 ,0)is removed from furtherconsideration since F (C 2 ,0) 0 , which in turnA2 B2 C2implies. As a geometric digression, any point(x,0)on the lower segment of BndD represents adegenerate case associated with Figure 2.33 since a viabletriangle cannot be generated if the y value (the altitude) iszero as depicted in Figure 2.35A A12xy = 0C-xA 3Figure 2.35: Carolyn’s Cauliflower for y = 0xcp[ C xTo examine the locus of critical pointscp] y2 cp0, we first need to ask the followingquestion: Given a critical point x , y ) on the locus with(cp cp0 x cp C , does the critical point necessarily lie in IntD ?Again, we turn to the techniques of single-variabledifferential calculus for an answer.75

Letxcp(cpbe the x component of an arbitrary critical pointxcp, y ) . Now xcpmust be such that x cp Cx [ C x ] cp0 0 inorder for the quantity 0 . This in turn allowstwo real-number values foronly thosecpcpy values where y 0ycp. In light of Figure 2.33,cpare of interest.(x cp, C) & G(C) < 0G(y) isdefined onthis line(x cp, y cp)&G(y cp) = 0(x cp, 0) & G(0) > 0Figure 2.36: Behavior of G on IntDDefine the continuous quadratic function2 x [ C x ] on the vertical line segmentG( y)ycpconnecting the two points ,0)cp( cpx and ( x cp, C)on BndD asshown in Figure 2.36. On the lower segment, we haveG ( 0) xcp[C xcp] 0 for all xcpin 0 x cp C . On theupper segment, we have that G(C) x [ C x ] C2 0for allx in x cp Ccp0 . This is since the maximum thatG( x,C)can achieve for any xcpin 0 x cp C is 3C2 / 4per the optimization techniques of single-variable calculus.Now, by the intermediate value theorem, there must be avalue0 y* C where G(y*) x [ C x ] ( y*)2 076cpcpcp cp.By inspection, the associated point ( x cp, y*)is in IntD ,and, thus, by definition is part of the locus of critical pointswith x , y*) ( x , y ) .(cpcp cp

Sincexcpwas chosen on an arbitrary basis, all criticalpoints , y )in IntD .3x defined by x [ C x ] y 0(cp cpcpcp2 cp : Thus far, we have used the techniques of differentialcalculus (both single and multi-variable) to establish alocus of critical points x [ C x ] y2 0 lying entirelycpwithin IntD . What is the relationship of each pointx , y ) to the Pythagorean Theorem? As observed at the(cp cpend of Step 1, one can immediately state the following:cpcplieF(xxAAmin22min, y22min[ C x B Bmin C) 0 ] y2 C22min 0 0 .M = (x cp, y cp)ηθγ = θ + ηLy cpα βx cpP C - x cpNFigure 2.37: D and Locus of Critical Points77

However, the last four-part equivalency is not enough. Weneed information about the three angles in our arbitrarytriangle generated via the critical point x , y ) as shownin Figure 2.37. In particular, is(cp cp a right angle? If2 2 2so, then the condition A B C corresponds to the factthat LMN is a right triangle and we are done! To proceed,first rewrite x [ C x ] y2 0 ascpcpcpxycpcpycp .C xcpNow study this proportional equality in light of Figure 2.37,where one sees that it establishes direct proportionality ofnon-hypotenuse sides for the two triangles LPMand MPN . From Figure 2.37, we see that both triangles haveinterior right angles, establishing thatThus LPM MPN . and . Since the sum of the remaining0two angles in a right triangle is 90 both00 90 . Combining 90 with the equality0 90 and and the definition for immediately leads to0 90 , establishing the key fact that LMN is aright triangle and the subsequent simultaneity of the twoconditionsA2 B2 C 2Figure 2.38 on the next page summarizes thevarious logic paths applicable to the now establishedCauliflower Proof of the Pythagorean Theorem with Converse.78

∂F/∂x = ∂F/∂y = 0x(C - x) 2 + y 2 = 0A 2 + B 2 = C 2F(x,y) = 0α + β = γFigure 2.38: Logically Equivalent Starting PointsStarting with any one of the five statements in Figure 2.38,any of the remaining four statements can be deduced viathe Cauliflower Proof by following a permissible path asindicated by the dashed lines. Each line is double-arrowedindicating total reversibility along that particular line.Once the Pythagorean Theorem is established, onecan show that the locus of pointsxcp[ C xcp] y2 cp0describes a circle centered at (C 2 ,0)with radiusC2 byrewriting the equation as2 2 2( xC2 ) [Ccp ycp 2] .This was not possible prior to the establishment of thePythagorean Theorem since the analytic equation for acircle is derived using the distance formula, a corollary ofthe Pythagorean Theorem. The dashed circle described by2 2 2( xC2 ) [Ccp ycp 2]in Figure 2.37 nicely reinforces the fact that LMN is aright triangle by the Inscribed Triangle Theorem of basicgeometry.79

Our final topic in this Section leads to a moregeneral statement of the Pythagorean Theorem.Let x , y ) be an arbitrary interior critical point as(cp cpestablished by the Cauliflower Proof. Except for the singlecboundary point ( 2,0)on the lower boundary of D ( f ) , theinterior critical points x , y ) are the only possible critical(cp cppoints. Create a vertical line segment passing throughas(cp cpx , y ) and joining the two points ( x ,0 cp) and ( , c)shown in Figure 2.39 below. As created, this vertical linesegment will pass through one and only one critical point.x cp(x cp, C)(0, C) (C, C)A1+A2>A3(x cp, y cp) & A1+A2=A3A1+A2

Define a new function21 A2 A3 2xcp 2cxcp 2g( x , y) Aycpfor all points y on the vertical line segment where the stickfigure is walking. The function g( x cp, y)is clearlycontinuous on the closed line segment y : y [0, c] wherey cp ( 0, c). We note the following three results:1. g( x ,0) 2x( x c) 0cpcpcp2cp2. g(x , y ) 2x[ c x ] y 0cpcp2cp3. g ( x , c) 2x 2c(c x ) 0 .cpcpBy continuity, we can immediately deduce that our functiong( x cp, y)has the following behavior directly associated withthe three results as given above:1. ( x , y) 0g cpfor all y 0, y )cpcp[cpThis in turn implies that A1 A2 A3.2. g( , ) 0x cpy cpThis in turn implies that A1 A2 A3.3. ( x , y) 0g cpfor all y ( y , c]This in turn implies that A1 A2 A3.Tying these results to the obvious measure of the associatedinterior vertex angle in Figure 2.37 leads to our finalrevised statement of the Pythagorean Theorem and converseon the next page.cp281

The Pythagorean Theorem andPythagorean ConverseSuppose we have a family of triangles built from acommon hypotenuse C with side lengths and angleslabeled in general fashion as shown below.BACThen the following three cases apply1.2.3. A A A222 B B B222 C C Cwith the Pythagorean Theorem and Converse being Case 2.22282

2.11) Shear, Shape, and AreaOur last major category of proof for the PythagoreanProposition is that of a shearing proof. Shearing proofs havebeen around for at least one thousand years, but they haveincreased in popularity with the advent of the moderncomputer and associated computer graphics.The heart of a shearing proof is a rectangle thatchanges to a parallelogram preserving area as shown inFigure 2.40. Area is preserved as long as the length andaltitude remain the same. In a sense, one could say that ashearing force F is needed to alter the shape of therectangle into the associated parallelogram, hence the nameshearing proof. Shearing proofs distinguish themselves fromtransformer proofs in that playing pieces will undergo bothshape changes and position changes. In transformer proofs,the playing pieces only undergo position changes.FbA1A2hFA3Identical b and hthroughout impliesA1=A2=A3Figure 2.40: Shearing a RectangleShearing proofs are primarily visual in nature, making themfantastic to watch when animated on a computer. As such,they are of special interest to mathematical hobbyists, whocollectively maintain their delightful pursuit of new ones.83

The one shearing proof that we will illustrate, without thebenefit of modern technology, starts with a variant ofEuclid’s Windmill, Figure 2.41. Four basic steps areneeded to move the total area of the big square into the twosmaller squares.Shear LineA1A2A1A21] Start2] Shear UpA1A2A2A13] Push Up 4] Shear OutFigure 2.41: A Four Step Shearing ProofStep descriptors are as follows:84

Step 1: Cut the big square into two pieces using Euclid’sperpendicular as a cutting guide.Step 2: Shear a first time, transforming the two rectanglesinto associated equal-area parallelograms whose slantedsides run parallel to the two doglegs of the master triangle.Step 3: Push the two parallelograms are pushed verticallyupward to the top of the shear line, which is a projection ofthe altitude for each square.Step 4: Shear a second time to squeeze both parallelogramsinto the associated little squares.To summarize the shear proof: The hypotenuse square is cutinto two rectangles per the Euclidean shear line. Therectangles are sheared into equivalent-area parallelograms,which are then pushed up the Euclidean shear line into aposition allowing each parallelogram to be sheared back tothe associated little square.85

2.12) A Challenge for All AgesAs shown in this chapter, proving the PythagoreanTheorem has provided many opportunities for mathematicaldiscovery for nearly 4000 years. Moreover, the PythagoreanTheorem does not cease in its ability to attract newgenerations of amateurs and professionals who want to addyet another proof to the long list of existing proofs. Proofscan be of many types and at many levels. Some are suitablefor children in elementary school such as the visual proofattributed to Pythagoras himself—super effective if madeinto a plastic or wood hand-manipulative set. Other proofsonly require background in formal geometry such as theshearing proof in Section 2.11. Still others requirebackground in both algebra and geometry. The CauliflowerProof requires a background in calculus. Thus, one can saythere is a proof for all ages as, indeed, there have beenproofs throughout the ages. The Pythagorean Crown Jewelnever ceases to awe and inspire!Type Example CommentVisualDissectionPythagoras Elementary SchoolAdvancedVisualLiu Hui Middle SchoolDissectionAlgebraicDissectionBhaskara Middle SchoolConstruction Euclid Early High SchoolTransformer Kurrah Middle SchoolSimilarity Legendre High SchoolTiling Perigal Early High SchoolCalculus ‘Cauliflower’ Early CollegeShearing Section 2.11 Early High SchoolTable 2.3: Categories of Pythagorean Proofs86

In Table 2.3, we briefly summarize the categories of existingproofs with an example for each. Not all proofs neatly tuckinto one category or another, but rather combine variouselements of several categories. A prime example isLegendre’s similarity proof. In Chapter 3, Diamonds of theSame Mind, we will explore a sampling of major mathematical‘spin offs’ attributed to the Pythagorean Theorem. In doingso, we will get a glimpse of how our Crown Jewel haspermeated many aspects of today’s mathematics and, bydoing so, underpinned many of our modern technologicalmarvels and discoveries.Euclid’s Beauty RevisitedNever did Euclid, as Newton ‘, discernAreas between an edge and a curveWhere ancient precisions of straight deferTo infinitesimal addends of turn,Precisely tallied in order to learnThose planes that Euclid could only observeAs beauty…then barren of quadratureAnd numbers for which Fair Order did yearn.Thus beauty of worth meant beauty in square,Or, those simple forms that covered the same.And, though, Archimedes reckoned with careArcs of exhaustion no purist would claim,Yet, his were the means for Newton to bareTrue Beauty posed…in Principia’s frame.October 200687

3) Diamonds of the Same Mind“Philosophy is written in this grand book—I mean the universe—whichstands continually open to our gaze, but it cannot be understood unlessone first learns to comprehend the language and interpret the charactersin which it is written. It is written in the language of mathematics, andits characters are triangles, circles, and other geometric figures, withoutwhich it is humanly impossible to understand a single word of it;without these, one is wandering about in a dark labyrinth.” GalileoGalilei3.1) Extension to Similar AreasAll squares are geometrically similar. When we ‘fit’ length—the primary dimension associated with a square—to all2 2 2three sides of a right triangle, we have that a b c orA A by the Pythagorean Theorem. The equationA1 2A31A2 A3 not only applies to areas of squares fitted tothe sides of a right triangle, but also to any geometricallysimilar planar figure fitted to the sides of a right trianglesuch as the three similar crosses shown in Figure 3.1.A1acbA2A1acbA2A3A3Figure 3.1: Three Squares, Three Crosses88

We can formally state this similarity principle as a theorem,one first espoused by Euclid for the special case of similarrectangles (Book 6, Proposition 31).Similar-Figure Theorem: Suppose three similar geometricfigures are fitted to the three sides of a right triangle insuch a fashion thatA ,21kaA 22kb, andA 23kcwhere the constant of proportionality k is identical for allthree figures. Then we have that A1 A2 A3.Proof: From the Pythagorean Theorema22 2 2 2 2 b c ka kb kc A1 A2 A3Table 3.1 provides several area formulas of the form2A kc for a sampling of geometric figures fitted to ahypotenuse of length c . In order to apply the similar-figuretheorem to any given set of three geometric figures, thefitting constant k must remain the same for the remainingtwo sides a and b .SIMILAR FIGUREDIMENSIONAREAFORMULAA 1cSquare Side length2RectangleSemicircleEquilateral TriangleLength or heightDiameterSide length2A chcA2 8cA 3 24c2Cross (Figure 3.1) Side length A 3c2Pentagon Side length A 1.72048cHexagonSide lengthA Table 3.1: A Sampling of Similar Areas3 3 22c89

3.2) Pythagorean Triples and Triangles53c m2 n2b m2 n24a 2mnFigure 3.2: Pythagorean TriplesA Pythagorean Triple is a set of three positiveintegers ( a,b,c)satisfying the classical Pythagoreanrelationshipa2 2 2 b c .Furthermore, a Pythagorean Triangle is simply a righttriangle having side lengths corresponding to the integers ina Pythagorean Triple. Figure 3.2 shows the earliest andsmallest such triple, ( 3,4,5), known to the Egyptians some4000 years ago in the context of a construction device forlaying out right angles (Section 2.1). The triple ( 6,8,10)wasalso used by early builders to lay out right triangles.Pythagorean triples were studied in their own right by theBabylonians and the Greeks. Even today, PythagoreanTriples are a continuing and wonderful treasure chest forprofessionals and amateurs alike as they explore variousnumerical relationships and oddities using the methods ofnumber theory, in particular Diophantine analysis (thestudy of algebraic equations whose answers can only bepositive integers). We will not even attempt to open thetreasure chest in this volume, but simply point the readerto the great and thorough references by Beiler andSierpinski where the treasures are revealed in full glory.90

What we will do in Section 3.2 is present theformulas for a well-known method for generatingPythagorean Triples. This method provides one of thetraditional starting points for further explorations ofPythagorean Triangles and Triples.Theorem: Let m n 0 be two positive integers. Then a setof Pythagorean Triples is given by the formula:( a,b,c) (2mn,m22 n , m2 n2)Proof:aaaa2222 b b b b22222 2 2 2 (2mn) ( m n ) 2 2 4 2 2 4mn m 2mn n4 2 2 4 m 2mn n 2 2 2 2 ( m n ) c 4From the formulas for the three Pythagorean Triples, wecan immediately develop expressions for both the perimeterP and area A of the associated Pythagorean Triangle.A 12(2mn)(mP 2mn m222 n ) mn(m n2 m291 n22 n2) 2m(m n)One example of an elementary exploration is to find allPythagorean Triangles whose area numerically equals theperimeter. To do so, first setA P mn(m2 nn(m n) 2 2) 2m(m n)m 3, n 1& m 3, n 2For m 3,n 1, we have that A P 24 .For m 3,n 2 , we have that A P 30 .

Table 3.2 lists all possible Pythagorean Triples with longside c 100 generated via the m & n formulas on theprevious page. Shaded are the two solutions where A P .M N A=2MN B=M 2 -N 2 C=M 2 +N 2 T P A2 1 4 3 5 PT 12 63 1 6 8 10 C 24 243 2 12 5 13 PT 30 304 1 8 15 17 P 40 604 2 16 12 20 C 48 964 3 24 7 25 PT 56 845 1 10 24 26 C 60 1205 2 20 21 29 P 70 210*5 3 30 16 34 C 80 2405 4 40 9 41 PT 90 1806 1 12 35 37 P 84 210*6 2 24 32 40 C 96 3846 3 36 27 45 C 108 4866 4 48 20 52 C 120 4806 5 60 11 61 PT 132 3307 1 14 48 50 C 112 3367 2 28 45 53 P 126 6307 3 42 40 58 C 140 840*7 4 56 33 65 P 154 9247 5 70 24 74 C 168 840*7 6 84 13 85 PT 182 5468 1 16 63 65 P 144 5048 2 32 60 68 C 160 9608 3 48 55 73 P 176 13208 4 64 48 80 C 192 15368 5 80 39 89 P 208 15609 1 18 80 82 C 180 7209 2 36 77 85 P 198 13869 3 54 72 90 C 216 19449 4 72 65 97 P 234 2340Table 3.2: Pythagorean Triples with c

There are three Pythagorean-triple types: primitive (P),primitive twin (PT), and composite (C). The definitions foreach are as follows.1. Primitive: A Pythagorean Triple ( a , b,c)where there is nocommon factor for all three positive integers a , b,&c .2. Primitive twin: A Pythagorean Triple ( a , b,c)where thelongest leg differs from the hypotenuse by one.3. Composite: A Pythagorean Triple ( a , b,c)where there is acommon factor for all three positive integers a , b,&c .The definition for primitive twin can help us find anassociated m & n condition for identifying the same. Ifa 2mn is the longest leg, thenc a 1mm22 n2 2mn n( m n)2 2mn12 1 1 m n 1Examining Table 3.2 confirms the last equality; primitivetwins occur whenever m n 1. As a quick exercise, weinvite the reader to confirm that the two cases m, n 8, 7and m , n 9, 8 also produce Pythagorean twins withc 100 .b m2 n2Ifis the longest leg [such as the case form 7 & n 2 ], thenc b 1m22n n221 m2 n21..93

The last equality has no solutions that are positive integers.Therefore, we can end our quest for Pythagorean Tripleswhere c b 1.Table 3.2 reveals two ways that CompositePythagorean Triples are formed. The first way is when thetwo generators m & n have factors in common, examples ofwhich are m 6 & n 2 , m 6 & n 3 , m 6 & n 4 orm 8 & n 4 . Suppose the two generators m & n have afactor in common which simply means m kp & m kq .Then the following is true:a 2mn 2kpkq 2kb mc m22 n n22 ( kp) ( kp)222pq ( kq) ( kq)22 k k22( p( p22 q q22)).The last expressions show that the three positive integersa , b & c have the same factor k in common, but now to thesecond power. The second way is when the two generatorsm & n differ by a factor of two: m n 2k,k 1,2,3....Exploring this further, we havea 2( n 2k)n 2nb ( n 2k)c ( n 2k)Each of the integers22 n n222 2k 2na b & c 2knn2 4k2 4nk 4n2, is divisible by two. Thus, allthree integers share, as a minimum, the common factortwo.In General: If k 0 is a positive integer and ( a , b,c)is aPythagorean triple, then ( ka , kb,kc)is also a PythagoreanTriple. Proof: left as a challenge to the reader..94

Thus, if we multiply any given primitive Pythagorean Triple,say ( 4,3,5), by successive integers k 2,3...; one can createan unlimited number composite Pythagorean Triples andassociated Triangles ( 8,6,10), ( 12,9,15), etc.We close this section by commenting on theasterisked * values in Table 3.2. These correspond to caseswhere two or more Pythagorean Triangles have identicalplanar areas. These equal-area Pythagorean Triangles aresomewhat rare and provide plenty of opportunity foramateurs to discover new pairs. Table 3.3 is a small tableof selected equal-area Pythagorean Triangles.A B C AREA20 21 29 21012 35 37 21042 40 58 84070 24 74 840112 15 113 840208 105 233 10920182 120 218 10920390 56 392 10920Table 3.3: Equal-Area Pythagorean TrianglesRarer yet are equal-perimeter Pythagorean Triangles.Table 3.4 shows one set of four equal-perimeterPythagorean Triangles where the perimeter P 1,000, 000 .Rumor has it that there are six other sets of four whereP 1,000,000 !A B C PERIM153868 9435 154157 31746099660 86099 131701 31746043660 133419 140381 31746013260 151811 152389 317460Table 3.4: Equal-Perimeter Pythagorean Triangles95

3.3) Inscribed Circle TheoremThe Inscribed Circle Theorem states that the radiusof a circle inscribed within a Pythagorean Triangle is apositive integer given that the three sides have beengenerated by the m & n process described in Section 3.2.Figure 3.3 shows the layout for the Inscribed CircleTheorem.Bb m2 n2ADrrrc ma 2mn2 n2CFigure 3.3: Inscribed Circle TheoremThe proof is simple once you see the needed dissection. Thekey is to equate the area of the big triangle ABC to thesum of the areas for the three smaller triangles ADB , BDC , and ADC . Analytic geometry deftly yields theresult.1 : Area(ABC)Area(ADB) Area(BDC) Area(ADC)2 : ()2mn(m1( )( r)2mn (2122rmn ( m22rmn 2m2212 n n)( r)(m2) ) r ( mr 2mn(m n n n) () m(n m)r mn(m n)(m n)r n(m n)222222212)( r)(m2) r 2mn(m n22) n2) 96

The last equality shows that the inscribed radius ris a product of two positive integral quantities n( m n).Thus the inscribed radius itself, called a PythagoreanRadius, is a positive integral quantity, which is what we setout to prove. The proof also gives us as a bonus the actualformula for finding the inscribed radius r n( m n).Table 3.5 gives r values for all possible m & n values wherem 7 .M N A=2MN B=M 2 -N 2 C=M 2 +N 2 R=N(M-N)2 1 4 3 5 13 1 6 8 10 23 2 12 5 13 24 1 8 15 17 34 2 16 12 20 44 3 24 7 25 35 1 10 24 26 45 2 20 21 29 65 3 30 16 34 65 4 40 9 41 46 1 12 35 37 56 2 24 32 40 86 3 36 27 45 96 4 48 20 52 86 5 60 11 61 57 1 14 48 50 67 2 28 45 53 107 3 42 40 58 127 4 56 33 65 127 5 70 24 74 107 6 84 13 85 6Table 3.5: Select Pythagorean Radii97

3.4) Adding a DimensionInscribe a right triangle in cattycorner fashion andwithin a rectangular solid having the three side lengths A,B, and C as shown in Figure 3.4. Once done, thePythagorean Theorem can be easily extended to threedimensions as follows.DCXBAFigure 3.4: Three Dimensional Pythagorean TheoremExtension Theorem: Let three mutually-perpendicular linesegments have lengths A, B and C respectively, and letthese line segments be joined in unbroken end-to-endfashion as shown in Figure 3.4. Then the square of thestraight-line distance from the beginning of the firstsegment to the end of the last line segment is given byA2 B2 C2 D2Proof: From the Pythagorean Theorem, we haveX2 C2 Dtime, we haveXAA2222 C B B. From the Pythagorean Theorem a secondA2222 B D X C2222 X& D22. Thus98

A Pythagorean Quartet is a set of four positiveintegers ( a,b,c,d)satisfying the three-dimensionalPythagorean relationshipa2 2 2 2 b c d .Pythagorean quartets can be generated using theformulas below.m & na m2b 2mnc 2nd m22 2n2To verify, square the expressions forsimplify.a b & c, , add, and( mm4( m2 (2mn) 4mn 4n2)22 2n)224 d2 (2n)22Table 3.6 lists the first twelve Pythagorean quartets sogenerated.M N A=M 2 B=2MN C=2N 2 D=M 2 +2N 22 1 4 4 2 63 1 9 6 2 113 2 9 10 8 174 1 16 8 2 184 2 16 16 8 244 3 16 24 18 345 1 25 10 2 275 2 25 20 8 335 3 25 30 18 435 4 25 40 32 576 1 36 12 2 386 2 36 24 8 44Table 3.6: Select Pythagorean Quartets99

Our last topic in Section 3.4 addresses the famousDistance Formula, a far-reaching result that is a directconsequence of the Pythagorean Theorem. The DistanceFormula—an alternate formulation of the PythagoreanTheorem in analytic-geometry form—permeates all of basicand advanced mathematics due to its fundamental utility.y( x2,y2,z2)zDC( x2 , y2, z1)B( x1,y1,z1)A( x2 , y1,z1)xFigure 3.5: Three-Dimensional Distance FormulaTheorem: Let ( x1,y1,z1)and ( x2,y2,z2) be two points inthree-dimensional Cartesian coordinate space as shown inFigure 3.5. Let D be the straight-line distance betweenthe two points. Then the distance D is given byProof:D ( xz .DD22D 2222 x1) ( y2 y1) ( z21)2 2 2 A B C 2 ( x x) ( y y)2( x21x)122 ( y21y)122 ( z2 ( z2z)112z)2100

3.5) Pythagoras and the Three MeansThe Pythagorean Theorem can be used to visuallydisplay three different statistical means or averages:arithmetic mean, geometric mean, and harmonic mean.Suppose a & b are two numbers. From these twonumbers, we can create three different means defined byArithmetic Mean:Geometric Mean:Harmonic Mean:a bM A2M G abMH1a21b2aba bCollectively, these three means are called the PythagoreanMeans due to their interlocking relationships with respectto six right-triangles as shown in Figure 3.6..ABRa GFyCxEbM AM GM HD GB R EC x FC yD a ba bR 2MHMGMAFigure 3.6: The Three Pythagorean Means101

In Figure 3.6, the line AD has length a b and isthe diameter for a circle with center [ a b]/ 2 . Triangle ACD utilizes AD as its hypotenuse and the circle rim asits vertex making ACDa right triangle by construction.Since EC is perpendicular to AD , ACE and ECD areboth similar to ACD . Likewise, the three triangles GCE , GFE , and EFC form a second similarity groupby construction.Next, we show that the three line segments GB ,EC , and FC are the three Pythagorean MeansandMHas defined.MA,MG,Arithmetic Mean: The line segment GB is a radius and haslength [ a b]/ 2 by definition. It immediately follows that:M Aa bGB 2 .Geometric Mean: The similarity relationship ACE ECD leads to the following proportion:xax2Mbx ab x G EC x ab ab.Harmonic Mean: By construction, the line segment GC is aradius with length[ a b]/ 2 . Line segment EF isconstructed perpendicular to GC . This in turn makes GCE EFC , which leads to the following proportionand expression for FC .102

yxy MHxab2ab2 y x FC y ab2ab 2ab y a b21a21bExamining the relative lengths of the three line segmentsGB , EC , and FC in Figure 3.6 immediately establishesthe inequality M M M for the three PythagoreanHMeans. The reader is invited to verify thatwhena b .GAMH MG MAs is usually done in mathematics, our two-numberPythagorean Mean definitions can easily be generalized tomulti-number definitions. LetThen the following definitions apply:General Arithmetic Mean:General Geometric Mean:General Harmonic Mean:a i, i 1,n be n numbers.MMMniA 1nnanGa ii1H nni1ai1 .To close, various sorts of means are some of themost widely used quantities in modern statistics. Three ofthese have geometric origins and can be interpreted withina Pythagorean context.iA103

3.6) The Theorems of Heron, Pappus,Kurrah, and StewartThe Pythagorean Theorem uses as its starting pointa right triangle. Suppose a triangle is not a right triangle.What can we say about relationships amongst the variousparts of a triangle’s anatomy: sides, areas, and altitudes?Most students of mathematics can recite the two basicformulas as presented in Figure 3.7. Are there other factsthat one can glean?Area 21chPerimeter a b cahcbFigure 3.7: The Two Basic Triangle FormulasIn this section, we will look at four investigations into thisvery question spanning a period of approximately 1700years. Three of the associated scientists andmathematicians—Heron, Pappus, and Stewart—are new toour study. We have previously introduced Kurrah in Section2.4.Heron, also known as Hero of Alexandria, was aphysicist, mathematician, and engineer who lived in the 1 stcentury ACE.Figure 3.8: Schematic of Hero’s Steam Engine104

Hero was the founder of the Higher Technical School at thethen world-renown center of learning at Alexandria, Egypt.He invented the first steam engine, schematically depictedin Figure 3.8, a device that rotated due to the thrustgenerated by steam exiting through two nozzlesdiametrically placed on opposite sides of the axis ofrotation. Working models of Hero’s steam engine are stillsold today as tabletop scientific collectables.In his role as an accomplished mathematician,Heron (as he is better known in mathematical circles)discovered one of the great results of classical mathematics,a general formula for triangular area known as Heron’sTheorem. Heron did not originally use the PythagoreanTheorem to prove his result. Both the Pythagorean Theoremand Heron’s Theorem are independent and co-equal resultsin that each can be derived without the use of the other andeach can be used to prove the other. In this section, we willshow how the Pythagorean Theorem, coupled with ‘heavy’analytic geometry, is used to render the truth of Heron’sTheorem. First, we state Heron’s historic result.Heron’s Theorem: Suppose a general triangle (a trianglehaving no special angles such as right angles) has threesides whose lengths are a , b , and c respectively. Lets [ a b c]/ 2 be the semi-perimeter for the same. Thenthe internal area A enclosed by the general triangle, Figure3.9, is given by the formula:A s( s a)(s b)(s c).ac xh xcbFigure 3.9: Diagram for Heron’s Theorem105

Proof of Heron’s Theorem:1 : Using the Pythagorean Theorem, create two equationsE1 & E 2in the two unknowns h and x .2E : hE122: h2 x : Subtract23x22cx ccx 2 b2 ( c x)2E from1 ( c x)222 b2 b2 b a2c a2 a2E and then solve for x .22 a2E . : Substitute the last value for x into14h2c : Solve for h .h h h h h 224cb24cb2 b a2c22222 c b24c2 c b24c b22 2 22 2 22cb c b a 2cb c b a 22ac b c b4c222 a aa b c a c b c b a c b a1062224c4c2222 a2

5 : Solve for area using the formulaA 21 ch .A A A 2 1ca b c a c b c b a c b aa b c a c b c b a c b a164 2ca b c 2c a b c 2b a c b 2a c b a22226 : Substitute s [ a b c]/ 2final result.and simplify to obtain theA 2c2b2as s s {s} 2 2 2 A ( s c)(s b)(s a)s A s(s a)(s b)(s c)***Pappus, like Heron, was a prominent mathematicianof the Alexandrian School and contributed much to the fieldof mathematics. There are several ‘Pappus Theorems’ still inuse today. Notable is the Pappus Theorem utilized in multivariablecalculus to determine the volume of a solid ofrevolution via the circular distance traveled by a centroid.In his masterwork, The Mathematical Collection (circa 300ACE), he published the following generalization of thePythagorean Theorem which goes by his name.Pappus’ Theorem: Let ABC be a general triangle withsides AB , AC , and BC . Suppose arbitrary parallelogramsA are fitted to sides AC and BC ashaving areas A1and2shown in Figure 3.10.107

If a parallelogram of area A3is fitted to side AB via theconstruction method in Figure 3.10, then A1 A2 A3.DdCA1A2AdEA 3BFA1 A2 A3Figure 3.10: Diagram for Pappus’ TheoremProof of Pappus’ Theorem:1 : Let ABC be a general triangle. Fit an arbitraryparallelogram of area A1to the side AC ; likewise, fit thearbitrary parallelogram of area A2to the side BC .2 : Linearly extend the outer sides—dotted lines—of thetwo fitted parallelograms until they meet at the point D.3 : Construct a dashed shear line through the points Dand C as shown in Figure 3.10. Let DC EF d .Construct two additional shear lines through points A andB with both lines parallel to the line passing through pointsD and C.108

4 : Construct the parallelogram with area3A as shownwith two sides of length AB and two sides of length DC .We claim that if the third parallelogram is constructed inA A .this fashion, then2 35 :1AFigure 3.11 depicts the two-step shearing proof forPappus’ Theorem. The proof is rather easy once therestrictions for the rectangle to be fitted on side AB areknown. Figure 3.11 shows the areas of the two originalparallelograms being preserved through a two-steptransformation that preserves equality of altitudes. This isdone by keeping each area contained within two pairs ofparallel guidelines—like railroad tracks. In this fashion,1is morphed to the left side of the constructed parallelogramhaving area A3. Likewise, A2is morphed to the right sideof the same. Summing the two areas gives the sought afterresult:A1 A2 A3AA A12A1A2A 3Figure 3.11: Pappus Triple Shear Line Proof109

Note 1 : The reader is invited to compare the shearing proof of Pappusto the one presented in Section 2.11 and notice the sequencereversal.Note 2 : Pappus’ Theorem is restrictive in that one must construct thethird parallelogram according to the process laid out in the proof.Perhaps the theorem evolved after many trials with the squares ofPythagoras as Pappus tried to relate size of squares to the sides ofa general triangle. In doing so, squares became rectangles andparallelograms as the investigation broadened. I suspect, as isusually the case, perspiration and inspiration combined to producethe magnificent result above!Like Herron’s Theorem, both the PythagoreanTheorem and Pappus’ Theorem are independent and coequalresults. Figure 3.12 shows the methods associatedwith Pappus’ Theorem as they are used to prove thePythagorean Theorem, providing another example of aPythagorean shearing proof in addition to the one shown inSection 2.11. The second set of equally spaced sheer lineson the left side, are used to cut the smaller square into twopieces so that a fitting within the rails can quite literallyoccur.Double ShearAnd Fit sideA2A1A1A2A3Figure 3.12: Pappus Meets Pythagoras***110

We already have introduced Thabit ibn Kurrah inSection 2.4 with respect to his Bride’s Chair and associatedtransformer proof. Kurrah also investigated non-righttriangles and was able to generalize the PythagoreanProposition as follows.Kurrah’s Theorem: Suppose a general triangle AED hasthree sides whose lengths are AD , AE , and ED . Let thevertex angle AED as shown in Figure 3.13.Construct the two line segments ED and EC in such afashion that both ABE and DCE . Then wehave thatAE22 ED AD(AB CD).EABCDFigure 3.13: Diagram for Kurrah’s TheoremProof of Kurrah’s Theorem:1 : Establish similarity for three key triangles by notingthat angle BAE is common to ABE and AED .Likewise EDC is common to ECD and AED . Thisand the fact that AEB ABE DCEimplies AED ABE ECD.111

2 :Set up and solve two proportional relationshipsAEABEDCDAD AEAEAD EDED22 AB AD & CD AD3 : Add the two results above and simplify to complete theproof.AEAE22 ED ED22 AB AD CD AD AD AB CDKurrah’s Theorem can be used to prove the Pythagorean0Theorem for the special case 90 via the following verysimple logic sequence.For, if the angle happens to be a right angle, then wehave a merging of the line segment EB with the linesegment EC . This in turn implies that the line segmentBC has zero length andKurrah’s result gives2AB CD AD2AE ED AD ,Note: Yet another proof of the Pythagorean Theorem!***2. Substituting intoMathew Stewart (1717-1785) was a professor at theUniversity of Edinburgh and a Fellow of the Royal Society.The famous theorem as stated on the next page—althoughattributed to Stewart—has geometric similarities totheorems originally discovered and proved by Apollonius.112

Some historians believe Stewart’s Theorem can be traced asfar back as Archimedes. Other historians believe thatSimpson (one of Stewart’s academic mentors) actuallyproved ‘Stewart’s Theorem’. Controversy aside, Stewart wasa brilliant geometer in his own right. Like Euclid, Stewartwas an expert organizer of known and useful results in theburgeoning new area of mathematical physics—especiallyorbital mechanics.Unlike the three previous theorems, Stewart’sTheorem is not co-equal to the Pythagorean Theorem. Thismeans that Stewart’s Theorem requires the use of thePythagorean Theorem in order to construct a proof.Independent proofs of Stewart’s Theorem are not possibleand, thus, any attempt to prove the Pythagorean Theoremusing Stewart’s Theorem becomes an exercise in circularlogic. We now proceed with the theorem statementmad hx nc m nbFigure 3.14: Diagram for Stewart’s TheoremStewart’s Theorem: Suppose a general triangle (atriangle having no special angles such as right angles) hassides with lengths a , b , and c . Let a line segment ofunknown length d be drawn from the vertex of the triangleto the horizontal side as shown in Figure 3.14, cutting cinto two portions m and n where c m n . Then, thefollowing relationship holds among the six line segmentswhose lengths arena2a , b,c,d,m , and n :2 2 mb c[d mn].113

Proof of Stewart’s Theorem1 : Using the Pythagorean Theorem, create three equationsE1 , E2,&E3in the three unknowns d , h , and x .E : bEE1232: a: d22 h2 h h22 ( n x)22 ( m x) x22 : Solve 3simplify.E for2h Substitute result in E1and E2,3E : bEE : bE12122: a2: a22 d d d22 d22 x x n222 m ( n x) ( m x) 2nx2 2mx : Formulate mE1 nE2and simplify using c m nmE : mbnEmb211222: na22mE nEmbmb na na na md nd22222: mb22 mn nm na22222 md 2mnx 2nmx nd mn2 ( m n)d mn(m n)2 cd mnc 2 c(d mn)22 .2 nm2Stewart’s result can be used to solve for the unknown linesegmentlength d in terms of the known lengths a , b,c,mand n .114

In pre-computer days, both all four theorems in this sectionhad major value to the practicing mathematician andscientist. Today, they may be more of an academiccuriosity. However, this in no way detracts from thosebefore us whose marvelous and clever reasoning led tothese four historic landmarks marking our continuingPythagorean journey.Table 3.7 summarizes the four theorems presentedin this section. A reversible proof in the context below is aproof that is can be independently derived and the used toprove the Pythagorean Theorem.WHO?WHAT?PROOFREVERSIBLE?PROOFSHOWN?Heron:GreekPappus:GreekKurrah:TurkishTriangular Area inTerms of SidesGeneralization forParallelogramsErected on sidesGeneralization forSquares of SideLengthsYesYesYesNoYesYesStewart:ScottishLength Formula forLine SegmentEmanating from ApexNoNotPossibleTable 3.7: Pythagorean Generalizations115

3.7) The Five Pillars of TrigonometryLanguage is an innate human activity, andmathematics can be defined as the language ofmeasurement! This definition makes perfect sense, forhumans have been both measuring and speaking/writingfor a very long time. Truly, the need to measure is in our‘blood’ just as much as the need to communicate.Trigonometry initially can be thought of as the mathematicsof ‘how far’, or ‘how wide’, or ‘how deep’. All of the precedingquestions concern measurement, in particular, themeasurement of distance. Hence, trigonometry, as originallyconceived by the ancients, is primarily the mathematicalscience of measuring distance. In modern times,trigonometry has been found to be useful in scores of otherapplications such as the mathematical modeling andsubsequent analysis of any reoccurring or cyclic pattern.Geometry, particularly right-triangle or Pythagoreangeometry, was the forerunner to modern trigonometry.Again, the ancients noticed that certain proportionsamongst the three sides of two similar triangles (triangleshaving equal interior angles) were preserved—no matter thesize difference between the two triangles. These proportionswere tabulated for various angles. They were then used tofigure out the dimensions of a large triangle using thedimensions of a smaller, similar triangle. This onetechnique alone allowed many powerful things to be donebefore the Common Era: e.g. construction of the GreatPyramid, measurement of the earth’s circumference (25,000miles in today’s terms), estimation of the distance from theearth to the moon, and the precise engineering of roadways,tunnels, and aqueducts. ‘Nascent trigonometry’, in the formof right-triangle geometry, was one of the backbones ofancient culture.In modern times trigonometry has grown far beyondits right triangle origins. It can now be additionallydescribed as the mathematics describing periodic or cyclicprocesses.116

One example of a periodic process is the time/distancebehavior of a piston in a gasoline engine as it repeats thesame motion pattern some 120,000 to 200,000 times in anormal hour of operation. Our human heart also exhibitsrepetitive, steady, and cyclic behavior when in good health.Thus, the heart and its beating motion can be analyzedand/or described using ideas from modern trigonometry ascan any electromagnetic wave form.Note: On a recent trip to Lincoln, Nebraska, I calculated that thewheel on our Toyota would revolve approximately 1,000,000 timesin a twelve-hour journey—very definitely a cyclic, repetitive process.Returning to measurement of distance, look up tothe night sky and think ‘how far to the stars’—much likeour technical ancestors did in ancient Greece, Rome,Babylon, etc. Trigonometry has answered that question inmodern times using the powerful parallax technique. Theparallax technique is a marriage of modern and old: careful,precise measurement of known distances/anglesextrapolated across vast regions of space to calculate thedistance to the stars. The Greeks, Romans, andBabylonians would have marveled! Now look down at yourGPS hand-held system. Turn it on, and, within a fewseconds, you will know your precise location on planetearth. This fabulous improvement on the compass operatesusing satellites, electronics, and basic trigonometry asdeveloped from right triangles and the associatedPythagorean Theorem. And if you do not have a GPS device,you surely have a cell phone. Every fascinating snippet ofcellular technology will have a mathematical foundation intrigonometry and the Pythagorean Theorem.Trigonometry rests on five pillars that areconstructed using direct Pythagorean principles or derivatesthereof. These five pillars serve as the foundation for thewhole study of trigonometry, and, from these pillars, onecan develop the subject in its entirety.Note: In 1970, while I was in graduate school, a mathematicsprofessor stated that he could teach everything that there is to knowabout trigonometry in two hours. I have long since realized that heis right. The five Pythagorean pillars make this statement so.117

To start a formal exploration of trigonometry, firstconstruct a unit circle of radius r 1and center ( 0,0). Lett be the counterclockwise arc length along the rim from thepoint ( 1,0)to the point P( t) ( x,y). Positive values for tcorrespond to rim distances measured in thecounterclockwise direction from ( 1,0)to P( t) ( x,y)whereas negative values for t correspond to rim distancesmeasured in the clockwise direction from ( 1,0)toP( t) ( x,y). Let be the angle subtended by the arclength t , and draw a right triangle with hypotenusespanning the length from ( 0,0)to ( x , y)as shown in Figure3.15.r 1P( t) ( x,y)ty( 0,0) x (1,0 )Figure 3.15: Trigonometry via Unit CircleThe first pillar is the collection definition for the sixtrigonometric functions. Each of the six functions has astheir independent or input variable either or t . Eachfunction has as the associated dependent or output variablesome combination of the two right-triangular sides whoselengths are x and y .118

The six trigonometric functions are defined as follows:1] ) sin( t) ysin( 2]csc( ) csc( t)1y3] ) cos( t) xcos( 4]1sec( ) sec( t)x5]ytan( ) tan( t) 6]xcot() cot( t)xyAs the independent variable t increases, one eventuallyreturns to the point ( 1,0)and circles around the rim asecond time, a third time, and so on. Additionally, any fixedpoint x , ) on the rim is passed over many times as we( y 0 0continuously spin around the circle in a positive or negativedirection. Let t0or 0be such thatr , one completeP( t0 0 0y0) P( ) ( x , ) . Since 1revolution around the rim of the circle is equivalent tounits and, consequently,P t ) P(t 2n ) : n 1,2,3... . We also have that(0 00( 0) P(3600) sin( 00t0 ) sin( t 2n) sin( 0) sin(360 0P . For )2t , this translates tosin( 0or ). Identicalbehavior is exhibited by the five remaining trigonometricfunctions.In general, trigonometric functions cycle through the samevalues over and over again as the independent variable tindefinitely increases on the interval [ 0, ) or, by the sametoken, indefinitely decreases on the interval (,0],corresponding to repeatedly revolving around the rim of theunit circle in Figure 3.15.119

This one characteristic alone suggests that (at least intheory) trigonometric functions can be used to model thebehavior of any natural or contrived phenomena having anoscillating or repeating action over time such as theinduced ground-wave motion of earthquakes or the inducedatmospheric-wave motion due to various types of music orvoice transmission.C( x,y)1yFigure 3.16: Trigonometry via General Right TriangleThe six trigonometric functions also can be definedusing a general right triangle. Let ABC be an arbitraryright triangle with angle as shown on the previous pagein Figure 3.16. The triangle ABC is similar totriangle ( 0,0),(1,0),( x,y)guarantees:1]3]5]( 0,0) x (1,0 )ABy BCACsin( ) 2] csc()1 AC 1y BCx ABACcos( ) 4] sec()1 AC 1x ABy BCx ABtan( ) 6] cot( ) x ABy AC. This right-triangle similarity120

The following four relationships are immediatelyevident from the above six definitions. These fourrelationships will comprise our primitive trigonometricidentities.1]3]1csc( ) 2]sin( )sin( )tan( ) 4]cos( )1sec( )cos( )cos( )cot( )sin( )Since the four trigonometric functions csc( ) , sec( ) ,tan( ) , and cot( ) reduce to nothing more thancombinations of sines and cosines, one only needs to knowthe value of sin( ) and )cos( for a given in order toevaluate these remaining four functions. In practice,trigonometry is accomplished by table look up. Tables forthe six trigonometric function values versus are set up00for angles from 0 to 90 in increments of 10 ' or less.Various trigonometric relationships due to embeddedsymmetry in the unit circle were used to producetrigonometric functional values corresponding to angles00ranging from 90 to 360 . Prior to 1970, these values weremanually created using a book of mathematical tables.Nowadays, the painstaking operation of table lookup istransparent to the user via the instantaneous modernelectronic calculator.Our second pillar is a group of three identities,collectively called the Pythagorean identities. ThePythagorean identities are a direct consequence of thePythagorean Theorem. From the Pythagorean Theorem andthe definitions of sin(t)and cos(t ) , we immediately seethat22sin ( t ) cos ( t) 1 .121

This is the first Pythagorean identity. Dividing the identity22sin ( t ) cos ( t) 1 , first by cos 2 ( t ) , and a second time bysin 2 ( t ) givestancot222( t)1 sec ( t)&2( t)1 csc ( t).These are the second and third Pythagorean identities. Allthree Pythagorean identities are extensively used intrigonometric analysis in order to convert from onefunctional form to another on an as-needed basis whendealing with various real-world problems involving anglesand measures of distances. In Section 4.3, we will explore afew of these fascinating real-world applications.The addition formulas for cos( ) , sin( )and tan( ) in terms of cos( ) , sin( ) , tan( ) ,cos( ) , and sin( ) , and tan( ) comprise as a group thethird pillar of trigonometry.{cos( ),sin( )}{cos( ),sin()} D 1( 0,0) (1,0 )D 2{cos( ), sin( )}Figure 3.17: The Cosine of the Sum122

As with any set of identities, the addition formulas provideimportant conversion tools for manipulation andtransforming trigonometric expressions into needed forms.Figure 3.17 on the previous page is our starting point fordeveloping the addition formula for the quantity, which statescos( )cos( ) cos( )cos( ) sin( )sin( ) .Using Figure 3.17, we develop the addition formula via fivesteps.1 : Since the angle in the first quadrant is equal tothe angle ( ) bridging the first and fourth quadrants,we have the distance equality2D1 D 2. : Use the Pythagorean Theorem (expressed in analyticgeometry form via the 2-D Distance Formula) to capture thisequality in algebraic language.D D12D[{cos( ),sin( )},(1,0)] D[{cos(),sin()},{cos( ), sin( )}] {cos( ) 1}{cos( ) cos( )}2{sin( ) 0}2{sin()[sin( )]}223 : Square both sides of the last expression:{cos( ) 1}{cos( ) cos( )}2{sin( ) 0}2{sin()[sin( )]}22123

4 : Square where indicated.522cos ( ) 2cos( ) 1sin ( ) 22cos ( ) 2cos( )cos( ) cos ( ) 22sin ( ) 2sin( )sin( ) sin ( )22 : Use the Pythagorean identity sin ( ) cos ( ) 1for any common angle , to simply to completion.2[cos ( ) sin2[cos ( ) cos[sin2( ) sin[1] 2cos( ) 1[1] 2cos( )cos( ) [1] 2sin( )sin( ) 2 2cos( ) 2 2cos( )cos( ) 2sin( )sin( ) 2cos( ) 2cos()cos( ) 2sin( )sin( ) 222( )] 2cos( ) 1( )] 2cos( )cos( ) ( )] 2sin( )sin( ) cos( ) cos( ) cos( ) sin( )sin( ) , trueThe last development, though lengthy, illustrates thecombined power of algebra and existing trigonometricidentities in order to produce new relationships for thetrigonometric functions. Replacing with and notingthatcos( ) cos( ) &sin( ) sin( )from Figure 3.17 immediately gives the companion formulacos( ) cos( )cos( ) sin( )sin( ) 124

Figure 3.17 can be used as a jumping-off point foran alternate method for simultaneously developing additionformulas for cos( ) and sin( ) . In Figure 3.18,the point {cos( ),sin( )} is decomposed intocomponents.y{cos( ),sin( )}x2 sin( )sin( )sin( )cos( )y2 sin( ) cos( ){cos( ),sin()}y1 sin( ) cos( )( 0,0) (1,0 )x1 cos( )cos( )xFigure 3.18: An Intricate TrigonometricDecompositionThis is done by using the fundamental definitions of sin( )and cos( ) based on both general right triangles and righttriangles having a hypotenuse of unit length as shown inFigure 3.16. The reader is asked to fill in the details on howthe various side-lengths in Figure 3.18 are obtained—agreat practice exercise for facilitating understanding ofelementary trigonometric concepts.125

From Figure 3.18, we have thatandcos( ) x1 x2cos( ) cos( )cos( ) sin( )sin( ) sin( ) y1 y2sin( ) sin( )cos( ) sin( )cos( )Using the trigonometric relationships cos( ) cos( ) andsin( ) sin( ) a second time quickly leads to thecompanion formula for sin( ) :sin( ) sin( )cos( ) sin( )cos( ).The addition formula for tan( )the addition formulas for sin( )following fashion is obtained from and cos( ) in thesin( )tan( ) cos( )sin( )cos( ) sin( )cos( )tan( ) cos( )cos( ) sin( )sin( )sin( )cos( ) sin( )cos( )cos( )cos( )tan( ) cos( )cos( ) sin( )sin( )cos( )cos( )tan( ) tan( )tan( ) 1tan( ) tan( )We will leave it to the reader to develop the companionaddition formula for tan( ) ..126

The remaining two pillars are the Law of Sines and the Lawof Cosines. Both laws are extensively used in surveyingwork and remote measuring of inaccessible distances,applications to be discussed in Section 4.3.CAaxx hy cybBFigure 3.19: Setup for the Law of Sines and CosinesFigure 3.19 is the setup diagram for both the Law of Sinesand the Law of Cosines. Let ABC be a general triangleand drop a perpendicular from the apex as shown. Then, forthe Law of Sines we have—by the fundamental definition ofsin( based on a general right triangle—thatsin( ) and )1h: sin( )bh bsin()2h: sin( ) ah asin( )3:bsin() asin( ) b a sin( ) sin( )127

The last equality is easily extended to include the thirdangle within ABC leading to our final result.Law of Sinesb a c sin( ) sin( )sin( )The ratio of the sine of the angle to the side opposite theangle remains constant within a general triangle.To develop the Law of Cosines, we proceed as followsusing the same triangle ABC as a starting point andrecalling that h bsin( ) .1 : Solve for y and x in terms of the angle 2y cos( ) y bcos()bx c y c bcos() : Use the Pythagorean Theorem to complete thedevelopment.x h[ c bcos()]cca2222 2bccos() b 2bccos() b c22 a b222[bsin()]22 a 2bccos( ) a2cos ( ) b2222sin2( ) aThe last equality is easily extended to include the thirdangle , leading to our final result on the next page.2128

a2Law of Cosines c2 b2 2bccos()The square of the side opposite the angle is equal to thesum of the squares of the two sides bounding the angleminus twice their product multiplied by the cosine of thebounded angle.Similar expressions can be written for the remaining twosides. We havecb22 a a22 b c22 2abcos( ) 2accos( )The Law of Cosines serves as a generalized form of thePythagorean Theorem. For if any one of the three angles(say in particular) is equal toimplying that.090 , then cos( ) 0aa22 c c22 b b22 2bc{0}In closing we will say that trigonometry in itself is avast topic that justifies its own course. This is indeed how itis taught throughout the world. A student is first given acourse in elementary trigonometry somewhere in highschool or early college. From there, advanced topics—suchas Fourier analysis of waveforms and transmissionphenomena—are introduced on an as-needed basisthroughout college and graduate school. Yet, no matter howadvanced either the subject of trigonometry or associatedapplications may become, all levels of modern trigonometrycan be directly traced back to the Pythagorean Theorem.129

3.8) Fermat’s Line in the SandChapters 1, 2, and 3 have turned the PythagoreanTheorem ‘every which way but loose’. For in these pages, wehave examined it from a variety of different aspects, provedit in a number of different ways, and developed some farreachingapplications and extensions. In order to provide afitting capstone, we close Chapter 4 with a mathematicalline in the sand, a world-renown result that not only statesthat which is impossible, but also subtlety implies that thePythagorean Theorem is the only example of the possible.Pierre de Fermat, a Frenchman, was born in thetown of Montauban in 1601. Fermat died in 1665. LikeHenry Perigal whom we met in Section 2.8, Fermat was adedicated amateur mathematician by passion, with his fulltimejob being that of a mid-level administrator for theFrench government. It was Fermat’s passion—over andabove his official societal role—that produced one of thegreatest mathematical conjectures of all time, Fermat’s LastTheorem. This theorem was to remain unproved until thetwo-year period 1993-1995 when Dr. Andrew Wiles, anEnglish mathematician working at Princeton, finally verifiedthe general result via collaboration with Dr. Richard N.Taylor of The University of California at Irvine.Note: A PBS special on Fermat’s Last Theorem revealed that Dr.Wiles had a life-long passion, starting at the age of 12, to be themathematician to prove Fermat’s Last Theorem. Dr. Wiles was in hisearly forties at the time of program taping. Wiles proof of Fermat’sLast Theorem is 121 pages long, which exceed the combined lengthof Chapters 2 through 4.So what exactly is Fermat’s Last Theorem? Before weanswer this question, we will provide a little more historicalbackground. A Diophantine equation is an algebra equationwhose potential solutions are integers and integers alone.They were first popularized and studied extensively by theGreek mathematician Diophantus (250 ACE) who wrote anancient text entitled Arithmetica.130

An example of a Diophantine equation is the Pythagorean2 2 2relationship x y z where the solutions x , y,z arerestricted to integers. We examined the problem of finding2 2 2integer solutions to x y z in Section 4.2 when westudied Pythagorean Triples. Likewise, Diophantus himselfexamined Pythagorean Triples and wrote about them in hisArithmetica. The subject of Pythagorean Triples was thefocus of Fermat’s studies sometime in 1637 (as the storygoes) when he pondered a natural extension.Fermat’s Fundamental QuestionSince many sets of three integers x , y,z exist wherex2 y2 z2, could it be that there exist a set of threeintegers x , y,z such thatx4 y4 z4, etc.x3 3 3 y z ? What aboutFor the cubic relationship, one can cometantalizingly close as the following five examples show:633 371 138135 833 3791 8122676 91383331 1443 3230 1721 1010333131 37533x31 y.3 zBut as we say in present times, almost does not countexcept when one is tossing horseshoes or hand grenades.Fermat must have quickly come to the realization that3 3 3 4 4 4x y z , x y z and companions constituted aseries of Diophantine impossibilities, for he formulated thefollowing in the margin of his copy of Arithmetica.3131

Fermat’s Last TheoremThe Diophantine equationn n nx y zwhere x , y , z , and n are all integers,has no nonzero solutions for n 2 .Fermat also claimed to have proof, but, alas, it wastoo large to fit in the margin of his copy of Arithmetica!Fermat’s Theorem and the tantalizing reference to a proofwere not to be discovered until after Fermat’s death in1665. Fermat’s son, Clement-Samuel, discovered hisfather’s work concerning Diophantine equations andpublished an edition of Arithmetica in 1670 annotated withhis father’s notes and observations.xn ynThe impossible integer relationship zn: n 2became known as Fermat’s LastTheorem—a theorem that could not be definitively proved ordisproved by counterexample for over 300 years until Wilesclosed this chapter of mathematical history in the two-yearspan 1993-1995. To be fair, according to current historians,Fermat probably had a proof for the case n 3 and the casen 4 . But a general proof for all n 2 was probablysomething way out of reach with even the bestmathematical knowledge available in Fermat’s day.***→132

However, Euler’s Conjecture did not stand the test oftime as Fermat’s Last Theorem did. In 1966, Lander andParker found a counterexample for n 5 :5 5 5 527 84 110133144Two counterexamples for n 4 followed in 1988. The firstwas discovered by Noam Elkies of Harvard. The second, thesmallest possible for a quartet of numbers raised to thefourth power, was discovered by Roger Frye of ThinkingMachines Corporation.4442,682,44015,465,639187,96044495,800 217,519 414,560 5422,48120,615,67344Today, power sums—both equal and non-equal—provide asource of mathematical recreation of serious and not-soseriousamateurs alike. A typical ‘challenge problem’ mightbe as follows:Find six positive integerssumsumuu77 7 7 v w x yv w x y zu , v,w,x,y,z satisfying the power7 z7where the associated linear is minimal.Table 3.7 on the next page displays just a few of theremarkable examples of the various types of power sums.With this table, we close Chapter 3 and our examination ofselect key spin-offs of the Pythagorean Theorem.→134

# EXPRESSION6 3 4 53 3371 3 7 13 3407 4 0 71 2135 1 3 51 2175 1 7 54 4 41634 1 6 3 43 4 33435 3 4 3 51 2598 5 9 85 5 5 554,748 5 4 7 4 83 3 3 312333435364758392169 13 and 961 3122025 45 and 20 25 4510211312 4913 17 and 4 9 13 171676 1 6 7 6 1 6 7 621233 12 332990100 990 100294122353 9412 23531 2 3 4 5 62646798 2 6 4 6 7 9 81 2 32427 2 4 2 73 3221859 22 18 593343 (3 4)1 2 3 4 5 4 3 213142152162177184193205Table 3.7: Power Sums135

4) Pearls of Fun and WonderPearls of ancient mind and wonder,Time will never pillage, plunder,Or give your soul to the worm—Or worse yet, for nerds to keepWith their mental treasures deepWhere secret squares are cut asunder.So, come good fun, have a turn,Bring your gold, build and learn!April 20054.1) Sam Lloyd’s Triangular LakeSam Lloyd was a famous American creator ofpuzzles, tricks and conundrums who produced most of hismasterpieces in the late 1800s. Many of Lloyd’s puzzleshave survived and actually thrived, having found their wayinto modern puzzle collections. Martin Gardner, of ScientificAmerican fame, has been a tremendous preserver of Lloyd’slegacy to recreational mathematics. Recreationalmathematics, one might classify that phrase as anoxymoron. Yet, back in the late 1800s, Lloyd’s heyday,people actually worked puzzles for evening relaxation, muchlike we moderns watch TV or play video games. The need torelax has always been there; how people fulfill the need ismore a function of the era in which we live and the availabletechnology that enables us to recreate.One of Sam Lloyd’s famous creations is hisTriangular Lake. Lloyd subtlety gives his readers twochoices: solution by sweat and brute force, or solution bycleverness and minimal effort. The clever solution requiresuse of the Pythagorean Theorem. What follows is Lloyd’soriginal statement:“The question I ask our puzzlists is to determine how manyacres there would be in that triangular lake, surrounded asshown in Figure 4.1 by square plots of 370, 116 and 74acres.136

The problem is of peculiar interest to those of a mathematicalturn, in that it gives a positive and definite answer to aproposition, which, according to usual methods produces oneof those ever-decreasing, but never-ending decimal fractions.TriangularLake370AcresAE54F74Acres 116AcresD 7 C 10BFigure 4.1: Triangle Lake and SolutionIn the year 2006, the Triangular Lake does notrequire a great deal of original thought to solve. Nowadays,we would say it is a ‘cookie cutter’ problem and requires nosweat whatsoever! One simply needs to apply Heron’sformula for triangular area, A s( s a)(s b)(s a)using a modern electronic calculator. The answer isavailable after a few keystrokes.1 : a c 2a b c : s 2s 19.304023 : A A 370 19.23538, b 74 8.60233370 116 2s(s a)(s b)(s a)116 10.77033,&7419.30402(0.06864)(8.53369)(10.70169) 11.00037137

We have achieved five-digit accuracy in ten seconds.Electronic calculators are indeed wonderful, but how wouldone obtain this answer before an age of technology?Granted, Heron’s formula was available in 1890, but theformula needed numbers to work, numbers phrased interms of decimal equivalents for those wary of roots. Toproduce these decimal equivalents would require themanual, laborious extraction of the three squareroots 370 , 116 , and 74 , not to mention extraction thefinal root in Heron’s Formula itself. Thus, a decimalapproach was probably not a very good fireside option in1890. One would have to inject a dose of cleverness, a lostart in today’s brute-force electronic world.Lloyd’s original solution entails a masterfuldecomposition of three right triangles. In Figure 4.1, thearea of the triangular lake is the area of ABF , which wedenote as .ABFANow: ABF ABD AFE FBC R EFCD .AAAAAFor ABDABAB, we have that22AB DA 922370 DB1722 370 Notice Lloyd was able to cleverly construct a right trianglewith two perpendicular sides (each of integral length)producing a hypotenuse of the needed length 370 . Thiswas not all he did!For AFEAFAF, we have that22 EA 522 7 EF22 74 AF 74138

Finally, for FBCFBFB22 CF 422 CB1022 116 FB 116Thus, our three triangles ABD , AFE , and FBC formthe boundary of the Triangular Lake. The area of theTriangular Lake directly followsAAABF ABF 12AABD AFE ( 9)(17) 112(5)(7) 2AAFBC RAEFCD (4)(10) (4)(7) 11The truth of Lloyd’s problem statement is now evident: tothose of a mathematical turn, the number 11 is a verypositive and definitive answer not sullied by an irrationaldecimal expansion.One might ask if it is possible to ‘grind through’Heron’s formula and arrive at 11 using the square roots asis. Obviously, due to the algebraic complexity, Lloyd wascounting on the puzzler to give up on this more brute-forcedirect approach and resort to some sort of cleverness.However, it is possible to grind! Below is the computationalsequence, an algebraic nightmare indeed.Note: I happen to agree with Lloyd’s ‘forcing to cleverness’ in that Ihave given students a similar computational exercise for years. Inthis exercise requiring logarithm use, electronic calculators aredeliberately rendered useless due to overflow or underflow ofderived numerical quantities. Students must resort to ‘old fashion’clever use of logarithms in order to complete the computationsinvolving extreme numbers..1 : a 370, b 2a b c : s 2116,& c 74370 116 274139

140212213116)74370116)(74370(370)74]116([41116)7437074)(370116(370)74116370)(74116(41))()((:AAasbsassA 1144419364132,40034,336411804(116)(74)41180)74116(2180)74116(241116)7411637011611674743707411637074370(370370)74741162(1164121212122121AAAA

4.2) Pythagorean Magic SquaresA magic square is an array of counting numbers(positive integers) geometrically arranged in a square.Figure 4.2 shows a 4X4 magic square. Where is the magicin this square?1 15 6 128 10 3 1311 5 16 214 4 9 7Figure 4.2: Pure and Perfect 4X4 Magic SquareAnswer: if one adds the four numbers in any one row, anyone column, or along any one of the two diagonals—totaling10 different ways—one will obtain the same number 34 foreach sum so done, called the magic sum.Normal or Pure Magic Squares are magic squares where thenumbers in the little squares are consecutive countingnumbers starting with one. Perfect 4X4 Magic Squares aremagic squares having many additional four-numberpatterns that sum to 34 , such as the four corners of anysmaller square embedded in the 4x4 square. Figure 4.3depicts a sampling of four-number patterns that sum to 34for the magic square shown in Figure 4.2.141

X X X X O X X O X XO X X O O OO X O X O OO O O O O X O X X XX X X X X O X OX X O O X O O XO O X X O X O XO O O O O X X OX X O O O X X XX X X X X O O OO O X X X O X XO O O O O X O OFigure 4.3: 4X4 Magic PatternsThe 4X4 magic square has within it several sum-ofsquaresand sum-of-cubes equalities that provide additionalexamples of power sums as discussed in Section 3.8.1. The sum of the squares in the first row equals the sumof the squares in the fourth row: A similar equalityholds for the second and third rows.2 2 2 2 2 2 21 15 6 12 14 4 9 2. The sum of the squares in the first column equals thesum of the squares in the fourth column. A similarequality holds for the second and third columns.2 2 2 2 2 2 21 8 11141213 2 3. The sum of the cubes for the numbers on the twodiagonals equals the sum of the cubes for the numbersremaining within the square:3 31 10153 63163133 73 2314 914233 5 433 33123 311 8The reader is invited to verify all three power sums!77322

Magic squares of all types have intrigued mathenthusiasts for decades. What you see below in Figure 4.4is a Pythagorean masterpiece that couples three magicsquares of different sizes (or orders) with the truth of thePythagorean Theorem. Royal Vale Heath, a well-knownBritish puzzle maker, created this wonder in England priorto 1930. Of the fifty numbers used in total, none appearsmore than once.THE PYTHAGOREAN 3-4-5 WONDER SET OFTHREE MAGIC SQUARES3X3 magicsum is 174.Square thesum of ninenumbers toobtain272,4845X5 magic sum is also4X4 magic sum is174. Square the sum ofalso 174. Squaretwenty-five numbers tothe sum of sixteenobtain 756,900numbers toobtain 484,416 16 22 28 34 7436 43 48 47 33 73 20 21 2761 54 59 49 46 37 42 25 26 32 72 1956 58 60 39 40 51 44 71 18 24 30 3157 62 55 50 45 38 41 29 35 70 17 233 2 2 2 4 5 & 272,484484,416 756, 900 !Figure 4.4: Pythagorean Magic Squares143

4.3) Earth, Moon, Sun, and StarsIn this section, we move away from recreational useof the Pythagorean Theorem and back to the physical worldand universe in which we live. In doing so, we will examinethe power of trigonometry as a tool to measure distances. Inparticular, we are interested in inaccessible or remotedistances, distances that we cannot ‘reach out and touch’ inorder to measure directly.The first example is determining the height H of theflagpole in front of the local high school. A plethora ofAmerican trigonometry students throughout the years havebeen sent outside to measure the height of the pole,obviously an inaccessible distance unless you entice thelittle guy to shimmy up the pole and drop a plumb bob. Becareful, for the principal may be looking!Figure 4.5 illustrates how we measured that oldschoolhouse flagpole using the elementary ideas oftrigonometry. We walked out a known distance from thebase of pole and sighted the angle from the horizontal to thetop of the pole.AH?5ft50 0 B=25ftFigure 4.5: The Schoolhouse FlagpoleNote: Students usually performed this sighting with a hand-madedevice consisting of a protractor and a pivoting soda straw. Inactual surveying, a sophisticated instrument called a theodolite isused to accomplish the same end.144

Once these two measurements are taken, the height of theflagpole easily follows if one applies the elementarydefinition of tan , as given in Section 3.7, to the righttriangle depicted in Figure 4.5 to obtain the unknownlength A.1Aft :25 tan(50A (25 ft) tan(50) ) A (25 ft)(1.19176) 29.79 ft2 : H 29.79 ft 5 ft 34.79 ft00A common mistake is the failure to add the height fromground level to the elevation of the sighting instrument.Figure 4.6 shows a marked increase in complexityover the previous example. Here the objective is to measurethe height H of a historic windmill that has been fenced offfrom visitors. To accomplish this measurement via thetechniques of trigonometry, two angular measurements aremade 25 feet apart resulting in two triangles and associatedbase angles as shown.AH?5f50 0B=25ft65 0 XFigure 4.6: Off-Limits Windmill145

The four-step solution follows. All linear measurements arein feet.1A0 : tan(50 ) 25 X0A (25 X ) tan(50 ) A (25 X ) (1.19176)A 29.79 1.19176X2 :AX tan(65) A 2.14451X X3A2.14451 : A 29.79 1.19176X A A 29.79 1.19176 2.14451A 29.79 .55573A0.44427A 29.79 A 67.05 ft.4 : H A 5 ft 72.05 ft 0Our next down-to-earth example is to find thestraight-line distance L through a patch of thorns andnettles—most definitely an inaccessible distance—as shownin Figure 4.7.A=100ft0 40B= 120ftL?Thorns andnettles.Figure 4.7: Across the Thorns and Nettles146

Finding the inaccessible distance L requires the measuringof the two accessible distances A & B and included angle .Once these measurements are obtained, the Law of Cosinesis directly utilized to find L.1 : LLL222 A 10022 B12022 2ABcos( ) 2(100)(120)cos(40 6014.93 L 77.556 ft ---0) Our next measurement, much more ambitious, wasfirst performed by Eratosthenes (275-194 BCE), theDirector of the Alexandrian Library in Egypt. Eratosthenesinvented an ingenious methodology for determining theearth’s circumference. His methodology utilized basictrigonometric principles in conjunction with threeunderlying assumptions quite advanced for his time:1) The earth was round2) When sunrays finally reached the earth aftertraveling across the unknown void, they arrived asparallel beams.3) Alexandria and the town of Syene (500 miles to theSouth) fell on the same meridian: an assumption notquite correct as shown in Figure 4.8.AlexandriaNileSyene(Aswan)Figure 4.8: Eratosthenes’ Egypt147

Figure 4.9 depicts Eratosthenes’ methodology. A mirrorwas placed at the bottom of a deep well at Syene. Thismirror would reflect back the rays of the noonday sun to anobserver during that time of year when the sun was shiningdirectly overhead. This was time correlated with a secondobservation at Alexandria where the shadow length of anobelisk (of know height) was measured allowingdetermination of the sun’s incident angle.7.2 0Parallel SunraysShadowObelisk atAlexandriaSunR & C=? Well atSyene7.2 0 Parallel SunraysMirrorFigure 4.9: Eratosthenes Measures the EarthEratosthenes reasoned that the circumferential distancefrom the obelisk at Alexandria to the well at Syene (5000miles) represented 7.2of the earth’s circumference. Asimple proportion was then solved allowing thedetermination of the earth’s total circumference C. Theearth’s radius R easily followed.1 O7.2 500miles : C 25,000milesO360 C225,000miles : R 3979miles2Eratosthenes came very close to modern measurements(The earth’s mean radius is 3959 miles ) by this ingeniousmethod now well over 2000 years old. It is a solid exampleof mathematics and natural science teaming to produce amilestone of rational thought in the history of humanity.148

Once we know the measurement of the earth’s radius, wecan immediately use the Pythagorean Theorem to calculatethe view distance V to an unobstructed horizon as afunction of the observer’s height H above the earth’ssurface, Figure 4.10.R+HVEarthRFigure 4.10: View Distance to Earth’s HorizonThe view-distance calculation proceeds as follows.VVVV222 R R22 ( R H ) R2 2RH H2RH H222 2RH HTable 4.1 gives select view distances to the horizon as afunction of viewer altitude.2H V H V5 ft. 2.73 miles 100,000 ft387.71miles100 ft 12.24 miles 150 miles 1100 miles1000ft 38.73 miles 1000 miles 2986 miles30,000 ft 212.18 miles R=3959 miles 3RTable 4.1: View Distance versus Altitude149

---High above the earth, we see the moon hanging inthe evening sky at an angle and inclination that variesaccording to latitude, date, and time. The ancient Greekswere the first to measure the distance to the moon and themoon’s diameter. They actually did this using severaldifferent ways. We will follow suit by employing the modernLaw of Sines, utilizing it one of the same geocentric setupsthat the Greeks used 2000 years ago in order to make thesetwo astronomical determinations. Today, our instrument ofchoice would be a theodolite in order to make the neededprecision angular measurements. The Greeks actually hadan ancient version of the same instrument, called anastrolabe, which allowed them to make the neededmeasurements in their day. Figure 4.11 shows our setupfor both measurements.Obelisk for measuring the Moon’sshadow in the absence of an astrolabeBF0.23620 014.47228 0 0.23620 0 Moon14.47228 0A 3959miles0.5D0Surface distanceEarthfrom B to D is1000milesCMoonFigure 4.11: Measuring the MoonIn order to measure the distance to the moon, firstpick two points B and D on the earth’s surface a knowndistance apart. Figure 4.11 suggests 1000 miles, roughlythe distance that the Greeks used. Both points need tohave either the same latitude or the same longitude. In theUnited States, two points of equal latitude (such as 40 0 N)are probably a tad easily to locate than two points of similarlongitude. Point D should correspond to a time of nightwhere the full moon is straight overhead or squarely in themiddle of the ecliptic as sighted by use of an astrolabe withthe sighting point squarely in the center of the moon’s disk.150

Simultaneously, an observer at point B would sight thecenter of the moon’s disk and ascertain the exterior angleFBCat shown. Modern email would make this an excitingexercise for amateur astronomers or high school studentssince these measurements can be communicatedinstantaneously. The ancient Greeks had no such luxury inthat they had to pre-agree as to a date and time that twoobservers would make the needed measurements. Severalweeks after that, information would be exchanged and thecalculations performed.By the Law of Sines, we can find the distance BC tothe moon per the following computational sequence.BC ABsin( BAD)sin( BCD)BCsin(14.47228 3959milesBC sin(0.23620BC 240,002.5miles3959miles0) sin(0.23620 )000sin(14.47228 ) )Notice that the ratio of the distance to the radius of theearth is given by the expressionBCAB 60.62,a traditional value first obtained by the Greeks after severalrefinements and iterations.Once we have the distance to the moon, we caneasily calculate the radius of the moon. Stand at point Dand measure the sweep angle between the moon’s two limbssighting through a diameter.151

The commonly accepted value is about 21degree, obtainedby holding out your thumb and, by doing so, barely0covering the lunar disk. Taking exactly half of 0.5tocomplete one very huge right triangle, we obtainRmoon0 tan(0.25 ) 240,002.5milesR 1047milesDmoonmoon 2Rmoon 2094milesThe calculated diameter 2094 miles is 66 miles shy of thetrue value. To illustrate the angular sensitivity associatedwith these results, suppose more sophisticatedinstrumentation indicates that the sweep angle between the0two lunar limbs is actually 0 .515 , an increase of 3 % .Revising the previous givesRmoon0 tan(0.2575 ) 240,002.5milesR 1078.63milesDmoonmoon 2157.26milesnow only a couple of miles shy of the true value. Whendealing with tiny angles, precision instrumentation is thekey to accurate results. However, the underlyingPythagorean infrastructure is still the same four millenniaafter its inception!---We will now use the distance to the moon to obtainthe distance to the sun. Like a series of celestial steppingstones,one inaccessible distance leads to a secondinaccessible distance, each succeeding distance moreextreme than the previous one.152

BACAB 87CAB 89.850: Greek0C: ModernFigure 4.12: From Moon to SunGreek astronomers knew that the moon reflected sunlightin order to shine. Using the cosmological model shown inABC 90Figure 4.12, the Greeks surmised thatduring the first quarter moon (half dark-half light). Byconstructing the huge imaginary triangle ABC , the moonto-sundistance BC could be obtained by the simpleformula0BCBA tan( BAC)assuming that the angle BAC could be accuratelymeasured. The Greeks tried and obtained a best value ofBAC 870, about six thumb widths away from thevertical. Modern instrumentation gives the value asBAC 89.850. The next computation is a comparison ofboth the ancient and modern values for the moon-to-sundistance using the two angular measurements in Figure4.12.Note: Staring directly into the sun is a very dangerous proposition,so the Greeks were quite naturally limited in their ability to refine‘six thumb widths’.153

1BC : tan( BAC)BABC BAtan(BAC)0BC (240,002.5) tan(87 ) BC 4,579,520.5miles20 : BC (240,002.5) tan(89.85 ) BC 91,673,992.7milesOur ‘back-of-the-envelope’ computed value has the sunabout 382 times further away from the earth than themoon. By contrast, the Greek estimate was about 19 times,a serious underestimate. However, even with anunderestimate, it is important to understand that Greekgeometry—again, Pythagorean geometry—was being usedcorrectly. The failing was not having sophisticatedinstrumentation. Knowing the distance to the sun, one caneasily obtain the sun’s diameter, whose disk also spans0approximately 0 .5 .Note: The sun and moon have the same apparent size in the skymaking solar eclipses possible.RSun0 tan(0.25 ) 91,673,992.7milesR 400,005.79milesDsunsun 800,011.58milesUsing Figure 4.11, we invite the reader to calculate the earth-tosundistance using the Pythagorean Theorem and the modern valueof the moon-to-sun distance shown above.---154

CEarth at point C in itsorbit about the sunSunBParallax angle> 1212AlphaCentauriDAEarth at point A, six months laterand diametrically opposite CFigure 4.13: From Sun to Alpha CentauriEarth, Moon, Sun, and Stars, the section title hints at aprogressive journey using ever-increasing stepping stones.The Greeks were limited to the known solar system.Starting in the 1800s, increasingly sophisticatedastronomical instrumentation made determination of stellardistances possible by allowing the measurement of very tinyangles just a few seconds in size, as depicted via the muchmagnified angle in Figure 4.13. From the measurementof tiny angles and the building of huge imaginaryinterstellar triangles, astronomers could ascertaintremendous distances using a Pythagorean-based methodcalled the parallax technique. Figure 4.13 illustrates theuse of the parallax technique to find the distance to ournearest stellar neighbor, Alpha Centauri, about 4 .2ly(lightyears) from the sun.The position of the target star, in this case Alpha Centari, ismeasured from two diametrically opposite points on theearth’s orbit. The difference in angular location against abackdrop of much farther ‘fixed’ stars is called the parallaxangle . Half the parallax angle is then used to compute thedistance to the star by the expression155

BD ACtan( 21 )Even for the closest star Alpha Centauri, parallax angles areextremely small. For AC 184,000, 000milesandBD 4. 2ly or ,463,592,320,000miles2 , we have1ACtan(2) BD1184,000,000tan(2) 0.00000747 2,463,592,320,00012 tan1 0.00085586(0.00000747) 0.000427930The final parallax value is less than one-thousandthof a degree. It converts to just 2.4 seconds. Being able todetermine angles this small and smaller is a testimony tothe accuracy of modern astronomical instrumentation. AsFigure 4.13 would hint, Smaller values of parallax anglesimply even greater distances. For example, a star whereBD 600ly (Betelgeuse in Orion is 522 ly from our sun)requires the measurement of a incredibly small parallax0angle whose value is 0.00000599 .The instantaneous elliptical diameter AC also needsprecision measurement in conjunction with the parallaxangle for a given star. In the preceding examples, werounded 91,673,992.7 to the nearest million and doubled,which resulted inAC 184,000, 000miles0. For precise workin a research or academic environment, AC would need tobe greatly refined.156

4.4) Phi, PI, and SpiralsOur last section briefly introduces three topics thatwill allow the reader to decide upon options for furtherreading and exploration. The first is the Golden Ratio, orPhi. Coequal to the Pythagorean Theorem in terms ofmathematical breadth and applicability to the naturalworld, the Golden Ratio deserves a book in its own right,and indeed several books have been written (seeReferences). In the discussion that follows, we brieflyintroduce the Golden Ratio and explore two instances wherethe Golden Ratio finds its way into right and non-righttriangles.ws ws wsh s wwFigure 4.14: The Golden RatioLet s be the semi-perimeter of a rectangle whosewidth and height are in the proportion shown in Figure4.14. This proportion defines the Golden Ratio, praised byartists and scientists alike. The equation in Figure 4.14reduces to a quadratic equation via the sequencesw ( s w)sw s22 2sw w2s2 3sw w2 0Solving (left to reader) gives the following values for thewidth and height:157

3w 2h s w 5 s w 0.38196s&5 1s w 0.61804s2 The Golden Ratio, symbolized by the Greek letter Phi ( ), isthe reciprocal of5 1 5 1, given by 1. 6180422.BBAababCA0363600360108a 72aa0C072bDFigure 4.15: Two Golden TrianglesAs stated, the Golden Ratio permeates mathematicsand science to the same extent as the PythagoreanTheorem. Figure 4.15 depicts two appearances in atriangular context, making them suitable inclusions for thisbook.TriangleABCthe vertical side has lengthon the left is a right triangle whereab equal to the geometricmean of the hypotenuse a and the horizontal side b .158

By the Pythagorean Theorem, we have:aa22 b2 ( ab bab) 0 15 a b 2 aa b b22This last result can be summarized as follows.If one short side of a right triangle is the geometric mean ofthe hypotenuse and the remaining short side, then the ratioof the hypotenuse to the remaining short side is the GoldenRatio .In triangle ABD to the right, all threetriangles ABD , ABC , and ACD are isosceles. Inaddition, the two triangles ABD and ACD are similar: ABD ACD . We have by proportionality rules:a b a a b22a ab b 0 a bThus, in Figure 4.15, ABD has been sectioned as tocreate the Golden Ratio between the slant height andbase for the two similar triangles ABC and ACD .Figure 4.16 aptly displays the inherent, unlabeled beautyof the Golden Ratio when applied to our two triangles.159

Figure 4.16: Triangular Phi---The history of PI, the fixed ratio of the circumferenceto the diameter for any circle, parallels the history of thePythagorean Theorem itself. PI is denoted by the Greekletter . Like the Golden Ratio Phi, PI is a topic of sufficientmathematical weight to warrant a complete book in its ownright. As one might suspect, books on PI have already beenwritten. In this volume, we will simply guide the reader tothe two excellent ‘PI works’ listed in the References.In this Section, we will illustrate just one of manydifferent ways of computing Pi to any desired degree ofaccuracy. We will do so by way of a nested iterativetechnique that hinges upon repeated use of the PythagoreanTheorem.Figure 4.17 on the next page shows a unit circlewhose circumference is given by C 2. Let T1be anisosceles right triangle inscribed in the first quadrant whosehypotenuse is h 12 . If we inscribe three right trianglescongruent to T1, one per each remaining quadrant, then acrude approximation to C is given by 4h 1 4 2 .160

T 2h 2h 12T 1h 3T 3C 2r&r 1Figure 4.17: Pythagorean PIBy bisecting the hypotenuse h1, we can create a smallerright triangle T2whose hypotenuse is h2. Eight of thesetriangles can be symmetrical arranged within the unit circleleading to a second approximation to C given by 8h2. Athird bisection leads to triangleT 3and a thirdapproximation 16h3. Since the ‘gap’ between the rim of thecircle and the hypotenuse of the right triangle generated byour bisection process noticeably tightens with successiveiterations, one might expect that the approximation for 2can be generated to any degree of accuracy, given enoughiterative cycles.Note: Analysis is the branch of mathematics addressing ‘endlessbehavior’, such as our bisection process above, which can go on adinfinitum. Some situations studied in analysis run counter tointuitively predicted behavior. Happily, the bisection process wasshown to converge (get as close as we like and stay there) to 2 inthe early 1990s.161

What remains to be done is to develop a formula forhi1that expresses hi1in terms of hi. Such a formula iscalled an iterative or recursive formula. Given a recursiveformula and the fact that h 12 , we should be able togenerate h2, then h3, and so on.1BA1h i2h i2AD 1CDh i1EFigure 4.18: Recursive HypotenusesFigure 4.18 shows the relationship between two successiveiterations and the associated hypotenuses. To develop ourrecursive formula, start with triangle ACE , which is aright triangle by construction since line segment AD bisectsthe angle BAE . We have by the Pythagorean Theorem:2 2AC CE AE22 hiAC1 AC2AC 4 h2 22i24 hi 4 42 (1)2162

Continuing, we form an expression for the side CDassociated with triangle CDE , also a right triangle.CD 1AC 2 CD 4 22h iAgain, by the Pythagorean Theorem2 2CD CE DE 2 hh2i1i14 h22 2 2i24 h44 h2i2i222 hi 2 h h2i2i1After a good bit of algebraic simplification (left to reader).The associated approximation for the actual circumferencei2 is given by the formula C i 2 1 hi. To establish theiterative pattern, we perform four cycles of numericcalculation.1 : h 121h 4 2 : h22, C2 1 224 h212 2C23 2 h2 82 2163

3 : h32 4 h222 2 24C 2 h33162 2 24 : h42 4 h232 2 2 2C45 2 h4 322 2 2 2In words, to calculate each succeeding hypotenuse, simply2 to the innermost 2 within the nestedadd one moreradicals. To calculate each succeeding circumferentialapproximation, simply multiply the associated hypotenuseby 2 to one additional power.ihici1 1.414 5.65692 0.7654 6.12293 0.3902 6.24294 0.1960 6.27315 0.0981 6.28076 0.0491 6.28267 0.0245 6.28308 0.0123 6.28319 0.0061 6.283210 0.0031 6.2832Table 4.2: Successive Approximations for 2Table 4.2 gives the results for the first ten iterations. Asone can see, the approximation has stabilized to the fourthdecimal place. Successive iterations would stabilizeadditional decimal places to the right of the decimal point.In this way, any degree of accuracy could be obtained if onehad enough time and patience. The true value of 2 tonine decimal places is 2 6. 283185307 .164

If we stabilize one digit for every two iterations (which seemsto be the indication by Table 4.2), then it would take abouttwenty iterations to stabilize our approximation to ninedigits—a great closing challenge to the reader!---Our last topic in Section 4.4 is that of PythagoreanSpirals. More art than mathematics, Pythagorean Spiralsare created by joining a succession of right triangles asshown in Figure 4.19. All nine triangles have outer sidesequal in length. In addition, the longer non-hypotenuse sideof a larger triangle is equal to the hypotenuse of thepreceding triangle. The generator for the ‘spiraling seashell’in Figure 4.19 is an isosceles right triangle of side lengthone. As one might imagine, the stopping point is arbitrary.Pythagorean Spirals make great objects for computergraphics programs to generate where coloration and precisealignment can be brought into play. Give it a try using moresophisticated software than the Microsoft Word utility that Iused to generate the spiraling seashell. Enjoy!11Figure 4.19: Pythagorean Spiral165

Epilogue: The Crown and the Jewels“Geometry has two great treasures;One is the Theorem of Pythagoras:The other, the division of a line into extreme and mean ratio.The first we may compare to a measure of gold;The second we may name a precious jewel.” Johannes KeplerSaint Paul says his First Epistle to the Corinthians,“For now we see through a glass, darkly…” Though thequote may be out of context, the thought freely standing onits own is carries the truth of the human condition.Humans are finite creatures limited by space, time, and theability of a three-pond mass to perceive the wondrousworkings of the Near Infinite…or Infinite. Even of that—Near Infinite or Infinite—we are not sure.Figure E.1: Beauty in Order166

Mathematics is one of the primary tools we humans use tosee through the glass darkly. With it, we can both discoverand describe ongoing order in the natural laws governingthe physical universe. Moreover, within that order we findcomfort, for order implies design and purpose—as opposedto chaos—through intelligence much grander than anythingproduced on a three-pound scale. The study of mathematicsas a study of order is one way to perceive cosmic-leveldesign and purpose. But no matter how good or howcomplete the human invention of mathematics can be, itstill allows just a glimpse through the darkened glass.Figure E.1, a mandala of sorts, hints at this truth oforder as it applies to the Pythagorean Theorem—which isbut a single pattern within the total synchronicity foundthroughout the cosmos. The four gradually shaded trianglesspeak of Pythagoras and one of the first Pythagorean proofs.Moreover, each triangle is golden via the ratio of hypotenuseto side, in deference to Kepler’s statement regarding the twogreat geometric treasures. Four also numbers the primarywaveforms in the visible spectrum—red, blue, green, andyellow— which can be mathematically modeled by advancedtrigonometric patterns (not addressed in this generalreadershipvolume). Since right triangles are intricatelylinked to circles, these two geometric figures are coupled inone-to-one fashion. A great circle, expressing unity,encompasses the pattern. White, as the synthesis of thevisual rainbow, appropriately colors all five circles. But evenwithin the beauty of pattern, linkage, and discovery asexpressed by our mandala, there is black. Let this darknessrepresent that which remains to be discovered or that whichis consigned to mystery. Either way, it becomes part of whowe are as humans as our innate finiteness seeks tocomprehend that which is much greater.It is my hope that this book has allowed a briefglimpse through the glass, a glimpse at a superbly simpleyet subtle geometric pattern called the PythagoreanTheorem that has endured for 4000 years or more ofhuman history.167

Not only has it endured, but the Pythagorean Theorem alsohas expanded in utility and application throughout thesame 40 centuries, paralleling human intellectual progressduring the same four millennia. In modern times, thePythagorean Theorem has found its long-standing trutheven more revered and revitalized as the theorem constantly“reinvents” itself in order to support new mathematicalconcepts.Today, our ancient friend is the foundation forseveral branches of mathematics supporting a plethora ofexciting applications—ranging from waveform analysis toexperimental statistics to interplanetary ballistics—totallyunheard of just two centuries ago. However, with a coupleof exceptions, the book in your hands has explored moretraditional trails. Like the color black in Figure E.1, there ismuch that you, the reader, can yet discover regarding thePythagorean Proposition. The eighteen works cited inAppendix D provide excellent roadmaps for furtherexplorations. We close with a classic geometric conundrum,Curry’s Paradox, to send you on your way: Given two sets offour identical playing pieces as shown in Figure E.2, howdid the square in the top figure disappear?Figure E.2: Curry’s Paradox168

AppendicesFigure A.0: The TangramNote: The Tangram is a popular puzzle currently marketedunder various names. Shown above in two configurations,the Tangram has Pythagorean origins.169

A] Greek AlphabetGREEK LETTERUpper CaseLower CaseΑ α AlphaΒ β BetaΓ γ GammaΔ δ DeltaΕ ε EpsilonΖ ζ ZetaΗ η EtaΘ θ ThetaΙ ι IotaΚ κ KappaΛ λ LambdaΜ μ MuΝ ν NuΞ ξ XiΟ ο OmicronΠ π PiΡ ρ RhoΣ σ SigmaΤ τ TauΥ υ UpsilonΦ φ PhiΧ χ ChiΨ ψ PsiΩ ω OmegaENGLISHNAME170

B] Mathematical SymbolsSYMBOLMEANINGPlus or Add- Minus or Subtract or Take AwayPlus or Minus: do both for two results / Divide · Multiply or Times{ } or[ ] or ( ) Parentheses1 , 2A BA BA BnIs equal toIs defined asDoes not equalIs approximately equal toIs similar tooIs greater thanIs greater than or equal toIs less thanIs less than or equal toStep 1, Step 2, etc.Implies the followingA implies BB implies AA implies B implies ASign for square rootSymbol for n th root|| ParallelPerpendicular Angle Right angleTriangle171

C] Geometric FoundationsThe Parallel Postulates1. Let a point reside outside a given line. Then there isexactly one line passing through the point parallel to thegiven line.2. Let a point reside outside a given line. Then there isexactly one line passing through the point perpendicularto the given line.3. Two lines both parallel to a third line are parallel toeach other.4. If a transverse line intersects two parallel lines, thencorresponding angles in the figures so formed arecongruent.5. If a transverse line intersects two lines and makescongruent, corresponding angles in the figures soformed, then the two original lines are parallel.Angles and Lines0 1800 901. Complimentary Angles: Two angles , with0 90 .172

2. Supplementary Angles: Two angles , with0 1803. Linear Sum of Angles: The sum of the two angles , formed when a straight line is intersected by a linesegment is equal to1804. Acute Angle: An angle less than00905. Right Angle: An angle exactly equal to6. Obtuse Angle: An angle greater thanTriangles900900ba 1800c1. Triangular Sum of Angles: The sum of the three interior0angles , , in any triangle is equal to 1802. Acute Triangle: A triangle where all three interiorangles , , are acute3. Right Triangle: A triangle where one interior angle from0the triad , , is equal to 904. Obtuse Triangle: A triangle where one interior angle0from the triad , , is greater than 905. Scalene Triangle: A triangle where no two of the threeside-lengths a , b,c are equal to another6. Isosceles Triangle: A triangle where exactly two of theside-lengths a , b,c are equal to each other7. Equilateral Triangle: A triangle where all three sidelengthsa b or all three angles ,a , b,c are identical c, are equal with 600173

8. Congruent Triangles: Two triangles are congruent(equal) if they have identical interior angles and sidelengths9. Similar Triangles: Two triangles are similar if they haveidentical interior angles10. Included Angle: The angle that is between two givensides11. Opposite Angle: The angle opposite a given side12. Included Side: The side that is between two given angles13. Opposite Side: The side opposite a given angleCongruent TrianglesGiven the congruent two triangles as shown belowbaedc1. Side-Angle-Side (SAS): If any two side-lengths and theincluded angle are identical, then the two triangles arecongruent.2. Angle-Side-Angle (ASA): If any two angles and theincluded side are identical, then the two triangles arecongruent.3. Side-Side-Side (SSS): If the three side-lengths areidentical, then the triangles are congruent.4. Three Attributes Identical: If any three attributes—sidelengthsand angles—are equal with at least one attributebeing a side-length, then the two triangles arecongruent. These other cases are of the form Angle-Angle-Side (AAS) or Side-Side-Angle (SSA).f174

Similar TrianglesGiven the two similar triangles as shown belowbae dc f1. Minimal Condition for Similarity: If any two angles areidentical (AA), then the triangles are similar.2. Ratio laws for Similar Triangles: Given similar trianglesas shown above, thenPlanar FiguresbecfadA is the planar area, P is the perimeter, n is the number ofsides.1. Degree Sum of Interior Angles in General Polygon:D 180 0 [ n 2]n 5 n 6n 5 D 540n 6 D 7200022. Square: A s : P 4s, s is the length of a sides175

3. Rectangle: A bh : P 2b 2h, b & h are the base andheightbh14. Triangle: A 2bh , b & h are the base and altitudebh5. Parallelogram: A bh , b & h are the base and altitude16. Trapezoid: A ( B b)h , B & b are the two parallelbases and h is the altitude227. Circle: A r: P 2rP d where d 2r, the diameter.hbbhB where r is the radius, orr8. Ellipse: A ab; a & b are the half lengths of the major& minor axesab176

D] ReferencesGeneral Historical Mathematics1. Ball, W. W. Rouse; A Short Account of the History ofMathematics; Macmillan &co. LTD., 1912; Reprintedby Sterling Publishing Company, Inc.,20012. Hogben, Lancelot; Mathematics for the Million; W.W. Norton & Company, 1993 Paperback Edition3. Polster, Burkard: Q.E.D. Beauty in MathematicalProof; Walker & Company, New York, 2004Pythagorean Theorem and Trigonometry4. Maor, Eli; The Pythagorean Theorem, a 4000-YearHistory; Princeton University Press; 20075. Loomis, Elisha; The Pythagorean Proposition;Publication of the National Council of Teachers;1968, 2 nd printing 1972; currently out of print6. Sierpinski, Waclaw; Pythagorean Triangles; DoverPublications; 20037. Lial, Hornsby, and Schneider; Trigonometry 7 th ;Addison-Wesley Educational Publishers Inc., 2001PI, Phi, and Magic Squares8. Beckman, Peter; A History of PI; The Golem Press,1971; Reprinted by Barnes & Noble, Inc., 19939. Posamentier, Alfred S. & Lehmann, Ingmar; PI, ABiography of the World’s Most Mysterious Number;Prometheus Books, New York, 200410. Livio, Mario; The Golden Ratio, the Story of PHI, theWorld’s Most Astonishing Number; Random HouseInc., New York, 2002177

Recreational Mathematics11. Pickover, Clifford A.; The Zen of Magic Squares,Circles, and Stars; Princeton University Press; 200212. Beiler, Albert H.; Recreations in the Theory ofNumbers, The Queen of Mathematics Entertains;Dover Publications; 196613. Pasles, Paul C.; Benjamin Franklin’s Numbers;Princeton University Press, 200814. Kordemsky, Boris A., The Moscow Puzzles; DoverPublications; 199215. Gardner, Martin; Mathematics Magic and Mystery;Dover Publications; 195616. Health Royal V.; Math-e-Magic: Math, Puzzles,Games with Numbers; Dover Publications; 1953Classical Geometry17. Euclid; Elements; Green Lion Press, 2002; updateddiagrams and explanations by D. Densmore and W.H. Donahue18. Apollonius; Conics Books I-IV (two volumes); GreenLion Press, 2000; revised reprint of 1939 translationby R. C. Taliaferro with updated diagrams andexplanations by D. Densmore and W. H. DonahueCalculus and Supporting Topics19. Landau, Edmund; Differential and Integral Calculus;Chelsea Publishing, 1980 Edition Reprinted by theAmerican Mathematic Society in 200120. Krauss, Eugene; Taxicab Geometry: an Adventure inNon-Euclidean Geometry; Dover Publications; 1987178

Topical IndexAAnalytic GeometryDefined 30Demonstrated on Pythagoras' proof 31-32AstronomyDistance from earth to moon 150-151Distance from moon to sun 152-154Distance from sun to Alpha Centauri 155-156Eratosthenes measures the earth 147-148Quote by Johannes Kepler 166Radius of the moon 151-152View distance to earth's horizon 148-149CCircleOne of three symmetric planar figures 17DDiophantine EquationDiscussed 131-133Distance FormulaDeveloped from Pythagorean Theorem 100Used to develop trig addition formulas 122-127EEuclidean MetricPythagorean example of 69Taxi-cab example of 69Rectangular example of 69Calculus based on other than Pythagorean 69-70Euler's ConjectureCounterexamples to 134Statement of 133FFermat's Last TheoremStatement of 132Andrew Wiles proved 130Richard Taylor's role in proving 130179

GGolden RatioDefined and algebraically developed 157-158Golden TrianglesDefined 159Two examples of 158-159HHeron's FormulaDeveloped from Pythagorean Theorem 105-107Used to solve Lloyd's Triangular Lake 137, 139-140Hero's Steam EngineDescribed with illustration 104-105IInscribed Circle TheoremPythagorean radius in context 97Reaffirmed by Cauliflower Proof 79Statement and proof 96KKurrah's TheoremStatement and proof 111-112Used to prove Pythagorean Theorem 112LLaw of CosinesDerived from Pythagorean Theorem 128-129Used in thorns-and-nettles problem 146-147Law of SinesDerived 127-128MMagic SquaresGeneral discussion 141-142MathematiciansAndrew Wiles 130Apollonius 36Archimedes 36, 47Bertrand Russell 36Bhaskara 23, 53-54Eratosthenes 147-148Euclid 35, 36, 84Galileo Galilei 88180

Mathematicians (cont)Henry Perigal 61-65Heron 104-107Johannes Kepler 166Legendre 41, 58-60Leonardo da Vinci 55-57Leonhard Euler 133Liu Hui 45-47Loomis, Elisha 16, 63, 68-70Mathew Stewart 112-113Pappus 107Pierre de Fermat 130-132President Garfield 66-77Pythagoras 27-35Richard Taylor 130Royal Vale Heath 143Sam Lloyd 136Thabit ibn Kurrah 49-52, 111Three greatest 47MeanArithmetic definition 101,103Geometric definition 101,103Harmonic definition 101,103PPappus' TheoremStatement and proof 107-109Used to prove Pythagorean Theorem 110PoetryCompared to prose 13-14Edna St. Vincent Millay 26"Euclid Alone Has Looked… " 26"Euclid's Beauty Revisited"© 87"From Earth to Love" © 3"Love Triangle" © 27"Pearls and Posers"© 136Relation to mathematics text 14"Significance" © 4Power SumsDiscussed 134-135PuzzlesGame Boy Boxel 47Curry's Paradox 168Kurrah's proof as source for 52181

Puzzles (cont)Pythagorean Magic Square by Royal Vale Heath 143Stomachion 47-48Tangram 169The Devil's Teeth © 52Transformers 49Triangular Lake by Sam Lloyd 136-140PythagoreanBook: The Pythagorean Proposition 16ConverseAs presented and proved by Euclid 42-44Generalized via Cauliflower Proof 80-83MeansDefinition 101-103Geometric interpretation 101-103Primal pattern conjectured 19QuartetsDefinition and formulas for 99RadiusDefinition and table 97SpiralsDefined and illustrated 165TilingFirst explained 62-65TriplesDefinition and properties 90-95Composite 93-95Formulas for generating 91Primitive 93-95Primitive twin 93-95Use of tem "perfection" with respect to 25-26Pythagorean TheoremAlgebraic square-within-a-square proof 32Archimedean origin speculated 47-48As "Crown Jewel" 86-87, 166-168As a Diophantine equation 132As known prior to Pythagoras 27-28As presented in Euclid's The Elements 36-37Sutton's embedded-similarity proof 59-60Basis for Pythagorean identities 121-122Bhaskara's algebraic-dissection proof 53-54Categories of Pythagorean proofs tabularized 86Cauliflower Proof 70-79Characterization of dissection proof 31-32182

Pythagorean Theorem (cont)Converse formally stated 42Converse proved per Euclid 43-44Distance formula as byproduct 100Earth-to-sun distance 154Egyptian 3-4-5 knotted rope 27-28Equivalency to other theorems tabularized 115Euclid's original Windmill Proof 37-39Euclid's Windmill Proof diagram first presented 1Euclid's Windmill Proof first mentioned 35Euler's Conjecture as spin off 133-134Extension to figures with similar areas 88Fermat's Last Theorem as spin off 130-133First proof by Pythagoras 29-30First proof by Pythagoras enhanced by algebra 31As foundation of trigonometry 116-118Garfield's trapezoidal dissection proof 66-67Generalized statement via Cauliflower Proof 80-82Henry Perigal's tiling proof 63-65Initial discovery speculated 23Initial statement of 24Kurrah's bride's-chair proof 49-51Legendre's embedded-similarity proof 58-59Leonardo da Vinci's asymmetric proof 55-57Liu Hui's packing proof 45-47Mosaic entitled "Pythagorean Dreams" 2Non-satisfaction by non-right triangles 2 5-26Power sums as spin off 134-135Pre-algebraic visual inspection proof 23-24Proved by Kurrah's Theorem 112Proved by Pappus' Theorem 110Pythagorean Magic Square 143Rectangular dissection proof 33Some proofs can be in several categories 87Speculative genesis 17-26Three-dimensional version and proof 98Twin-triangle dissection proof 34-36Used to calculate PI 160-164Used to define golden triangles 158-160Used to derive Law of Cosines 128-129Used to develop Heron's Formula 105-107Used to develop trigonometric addition formulas 122-126With respect to amateur mathematicians 13With respect to modern technology 13Windmill-light proof using analytic geometry 40-41183

Pythagorean Theorem ProofsAlgebraicFirst explained 53-54Bhaskara's algebraic dissection proof 53-54CalculusCauliflower Proof 70-79Cauliflower Proof logic links 79First explained 68-69Diagram, Carolyn's Cauliflower 70-71, 75Elisha Loomis states impossibility of 68-69ConstructionFirst explained 37Windmill using analytic geometry 40-41Euclid's original Windmill Proof 37-39Leonardo da Vinci's asymmetric proof 55-58DissectionBride's Chair traditional classification 49Dissection Order (DR) system explained 32Garfield's trapezoidal dissection proof 66-67Square-within-a-square proof 31Triangles-within-a-rectangle proof 33Twin-triangle proof by Watkins 34-35PackingConjectured origin of Liu Hui's proof 47-48First explained 46-47ShearingFirst explained 83Sutton's embedded-similarity Proof 59-61Euclid's Windmill and shearing proofs 84SimilarityFirst explained 41Illustrated 84-85Legendre's embedded-similarity proof 58-59Exploited in windmill-light proof 41TilingFirst explained 63-64Henry Perigal's tiling proof 63-66Infinite number of 63TransformerAlternate classification of bride's-chair 49Kurrah's "Operation Transformation" 50-51184

QQuadraticEquation 14-15Formula 14-15RRectanglePartitioned into right triangles via a diagonal 21Four used to make square with square hole 22SSimilar Figure TheoremConstructed on sides of right triangle 88-89SquareFour suggested planar actions/movements 18One of three symmetric planar figures 17Partitioned and subsequently replicated 19Partitioned into right triangles via a diagonal 18-19Stewart's TheoremStatement and proof 113-114Not equivalent to Pythagorean Theorem 113Requires Pythagorean Theorem to prove 113TTombstoneHenry Perigal's 61-62TriangleEquilateralOne of three symmetric planar figures 17PythagoreanDefinition 90Equal area 95Equal perimeter 95Area numerically equal to perimeter 91RightDefinition and angular properties 20-21Formed from rectangle via a diagonal 21Hypotenuse 22Right IsoscelesCreated from square via a diagonal 18-19TrigonometryAddition formulas 122-126Distance from earth to moon 150-152Distance from moon to sun 152-154Distance from sun to Alpha Centauri 155-157185

Trigonometry (cont)Eratosthenes measures the earth 147-148Law of Cosines 128-129Law of Sines 127-128PI calculated using Pythagorean Theorem 160-164Primitive identities 121Pythagorean identities 121-122Radius of the moon 151-153Right-triangle definitions of trig functions 120-121Schoolhouse flagpole problem 144-145Thorns-and-nettles problem 146-147Unit-circle definitions of trig functions 118-119View distance to earth's horizon 148-150186

About the Author John C. Sparks is a retiredengineer at Wright-Patterson AirForce Base (Dayton, Ohio) with over33 years of aerospace experience.Additionally, Sparks is an activeAdjunct Instructor of mathematicsat Sinclair Community College(Dayton) with over 25 years ofteaching experience. In late 2003,Sparks received the OhioAssociation of Two-Year Colleges’2002-2003 Adjunct Teacher of theYear award. He and his wife,Carolyn (photo), have celebrated 39years of marriage and have twogrown sons, Robert and Curtis.Sparks is a lifelong resident ofXenia, Ohio.In addition to The Pythagorean Theorem, Sparks haspublished three other books: Calculus without Limits 3 rd , abeginner’s primer to single-variable calculus; Gold, Hay andStubble, a book of original poems which muses the time period1996 to 2006 in diary format; and, per Air Force sponsorship, TheHandbook of Essential Mathematics, an e-book available as a freePDF download on http://www.asccxe.wpafb.af.mil.

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