The Pythagorean Theorem - Educational Outreach

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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
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The PythagoreanTheoremCrown Jewel o
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The Pythagorean TheoremCrown Jewel
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Table of ContentsList of Tables and
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List of Tables and FiguresTablesNum
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Figures…continuedNumber and Title
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List of Proofs and DevelopmentsSect
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PrefaceThe Pythagorean Theorem has
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1 : 2x2x2****22 13x 7 13x 7 0 : a
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1) Consider the Squares“If it was
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s090045 0ss45450450090Figure 1.3: O
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Notice that the right-triangle prop
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The middle square (minus the donut
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Our proof in Chapter 1 has been by
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2) Four Thousand Years of Discovery
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The proof Pythagoras is thought to
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cabcFigure 2.4: Annotated Square wi
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Figure 2.6 is the diagram for a sec
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1x:b2:Area(CBA)3:Area( ABC)Area(ABC
Page 39 and 40: Euclid’s proof of the Pythagorean
Page 41 and 42: First we establish that the two tri
Page 43 and 44: 1 2y b b: y b c c2:A3:A4:Aunshaded
Page 45 and 46: XB'A090 CBFigure 2.12: Euclid’s C
Page 47 and 48: 2.3) Liu Hui Packs the SquaresLiu H
Page 49 and 50: One of my favorites was Boxel, a ga
Page 51 and 52: 2.4) Kurrah Transforms the Bride’
Page 53 and 54: Note: as is the occasional custom i
Page 55 and 56: 2.5) Bhaskara Unleashes the Power o
Page 57 and 58: 2.6) Leonardo da Vinci’s Magnific
Page 59 and 60: In Figure 2.24, we enlarge Step 5 a
Page 61 and 62: But, this is precisely the nature o
Page 63 and 64: 2.8) Henry Perigal’s TombstoneHen
Page 65 and 66: In the century following Perigal, b
Page 67 and 68: Returning to Henry Perigal, Figure
Page 69 and 70: Figure 2.32 is President Garfield
Page 71 and 72: Even though the Cartesian coordinat
Page 73 and 74: B 2← γ →α y βx C-xC 2Figure
Page 75 and 76: Being polynomial in form, the funct
Page 77 and 78: Thus, the critical point ( C 2 ,0)i
Page 79 and 80: Sincexcpwas chosen on an arbitrary
Page 81 and 82: ∂F/∂x = ∂F/∂y = 0x(C - x) 2
Page 83 and 84: Define a new function21 A2 A3 2xcp
Page 85 and 86: 2.11) Shear, Shape, and AreaOur las
Page 87 and 88: Step 1: Cut the big square into two
Page 89: In Table 2.3, we briefly summarize
Page 93 and 94: What we will do in Section 3.2 is p
Page 95 and 96: There are three Pythagorean-triple
Page 97 and 98: Thus, if we multiply any given prim
Page 99 and 100: The last equality shows that the in
Page 101 and 102: A Pythagorean Quartet is a set of f
Page 103 and 104: 3.5) Pythagoras and the Three Means
Page 105 and 106: yxy MHxab2ab2 y x FC y ab2ab 2ab y
Page 107 and 108: Hero was the founder of the Higher
Page 109 and 110: 5 : Solve for area using the formul
Page 111 and 112: 4 : Construct the parallelogram wit
Page 113 and 114: We already have introduced Thabit i
Page 115 and 116: Some historians believe Stewart’s
Page 117 and 118: In pre-computer days, both all four
Page 119 and 120: One example of a periodic process i
Page 121 and 122: The six trigonometric functions are
Page 123 and 124: The following four relationships ar
Page 125 and 126: As with any set of identities, the
Page 127 and 128: Figure 3.17 can be used as a jumpin
Page 129 and 130: The remaining two pillars are the L
Page 131 and 132: a2Law of Cosines c2 b2 2bccos()The
Page 133 and 134: An example of a Diophantine equatio
Page 136 and 137: However, Euler’s Conjecture did n
Page 138 and 139: 4) Pearls of Fun and WonderPearls o
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We have achieved five-digit accurac
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140212213116)74370116)(74370(370)74
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X X X X O X X O X XO X X O O OO X O
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4.3) Earth, Moon, Sun, and StarsIn
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The four-step solution follows. All
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Figure 4.9 depicts Eratosthenes’
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---High above the earth, we see the
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The commonly accepted value is abou
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1BC : tan( BAC)BABC BAtan(BAC)0BC
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BD ACtan( 21 )Even for the closest
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3w 2h s w 5 s w 0.38196s&5 1s
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Figure 4.16: Triangular Phi---The h
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What remains to be done is to devel
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3 : h32 4 h222 2 24C 2 h33162 2 2
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Epilogue: The Crown and the Jewels
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Not only has it endured, but the Py
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A] Greek AlphabetGREEK LETTERUpper
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C] Geometric FoundationsThe Paralle
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8. Congruent Triangles: Two triangl
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3. Rectangle: A bh : P 2b 2h, b &
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Recreational Mathematics11. Pickove
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GGolden RatioDefined and algebraica
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Puzzles (cont)Pythagorean Magic Squ
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Pythagorean Theorem ProofsAlgebraic
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Trigonometry (cont)Eratosthenes mea
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