The Pythagorean Theorem - Educational Outreach

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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
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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
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XB'A090 CBFigure 2.12: Euclid’s C
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2.3) Liu Hui Packs the SquaresLiu H
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One of my favorites was Boxel, a ga
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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 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: 3.5) Pythagoras and the Three Means
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
Page 146 and 147: 4.3) Earth, Moon, Sun, and StarsIn
Page 148 and 149: The four-step solution follows. All
Page 150 and 151: Figure 4.9 depicts Eratosthenes’
Page 152 and 153: ---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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