The Pythagorean Theorem - Educational Outreach

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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
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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
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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 184 and 185: Puzzles (cont)Pythagorean Magic Squ
Page 188: Trigonometry (cont)Eratosthenes mea
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The Pythagorean Theorem - Educational Outreach

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