The Pythagorean Theorem - Educational Outreach

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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
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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’
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Note: as is the occasional custom i
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2.5) Bhaskara Unleashes the Power o
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2.6) Leonardo da Vinci’s Magnific
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In Figure 2.24, we enlarge Step 5 a
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But, this is precisely the nature o
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2.8) Henry Perigal’s TombstoneHen
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In the century following Perigal, b
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Returning to Henry Perigal, Figure
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Figure 2.32 is President Garfield
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Even though the Cartesian coordinat
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B 2← γ →α y βx C-xC 2Figure
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Being polynomial in form, the funct
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Thus, the critical point ( C 2 ,0)i
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Sincexcpwas chosen on an arbitrary
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∂F/∂x = ∂F/∂y = 0x(C - x) 2
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Define a new function21 A2 A3 2xcp
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2.11) Shear, Shape, and AreaOur las
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Step 1: Cut the big square into two
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In Table 2.3, we briefly summarize
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We can formally state this similari
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What we will do in Section 3.2 is p
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There are three Pythagorean-triple
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Thus, if we multiply any given prim
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The last equality shows that the in
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A Pythagorean Quartet is a set of f
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3.5) Pythagoras and the Three Means
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yxy MHxab2ab2 y x FC y ab2ab 2ab y
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Hero was the founder of the Higher
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5 : Solve for area using the formul
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4 : Construct the parallelogram wit
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We already have introduced Thabit i
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Some historians believe Stewart’s
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In pre-computer days, both all four
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One example of a periodic process i
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The six trigonometric functions are
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The following four relationships ar
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As with any set of identities, the
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Figure 3.17 can be used as a jumpin
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The remaining two pillars are the L
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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
Page 154 and 155: The commonly accepted value is abou
Page 156 and 157: 1BC : tan( BAC)BABC BAtan(BAC)0BC
Page 158 and 159: BD ACtan( 21 )Even for the closest
Page 160 and 161: 3w 2h s w 5 s w 0.38196s&5 1s
Page 162 and 163: Figure 4.16: Triangular Phi---The h
Page 164 and 165: What remains to be done is to devel
Page 166 and 167: 3 : h32 4 h222 2 24C 2 h33162 2 2
Page 168 and 169: Epilogue: The Crown and the Jewels
Page 170 and 171: Not only has it endured, but the Py
Page 172 and 173: A] Greek AlphabetGREEK LETTERUpper
Page 174 and 175: C] Geometric FoundationsThe Paralle
Page 176 and 177: 8. Congruent Triangles: Two triangl
Page 178 and 179: 3. Rectangle: A bh : P 2b 2h, b &
Page 180 and 181: Recreational Mathematics11. Pickove
Page 182 and 183: GGolden RatioDefined and algebraica
Page 186 and 187: Pythagorean Theorem ProofsAlgebraic
Page 188: Trigonometry (cont)Eratosthenes mea
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