The Pythagorean Theorem - Educational Outreach

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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
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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
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Euclid’s proof of the Pythagorean
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First we establish that the two tri
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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 and 90: In Table 2.3, we briefly summarize
Page 91 and 92: We can formally state this similari
Page 93 and 94: What we will do in Section 3.2 is p
Page 95: There are three Pythagorean-triple
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
Page 140 and 141: We have achieved five-digit accurac
Page 142 and 143: 140212213116)74370116)(74370(370)74
Page 144 and 145: 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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The Pythagorean Theorem - Educational Outreach

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