The Pythagorean Theorem - Educational Outreach

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GGolden RatioDefined and algebraically developed 157-158Golden TrianglesDefined 159Two examples of 158-159HHeron's FormulaDeveloped from Pythagorean Theorem 105-107Used to solve Lloyd's Triangular Lake 137, 139-140Hero's Steam EngineDescribed with illustration 104-105IInscribed Circle TheoremPythagorean radius in context 97Reaffirmed by Cauliflower Proof 79Statement and proof 96KKurrah's TheoremStatement and proof 111-112Used to prove Pythagorean Theorem 112LLaw of CosinesDerived from Pythagorean Theorem 128-129Used in thorns-and-nettles problem 146-147Law of SinesDerived 127-128MMagic SquaresGeneral discussion 141-142MathematiciansAndrew Wiles 130Apollonius 36Archimedes 36, 47Bertrand Russell 36Bhaskara 23, 53-54Eratosthenes 147-148Euclid 35, 36, 84Galileo Galilei 88180
Mathematicians (cont)Henry Perigal 61-65Heron 104-107Johannes Kepler 166Legendre 41, 58-60Leonardo da Vinci 55-57Leonhard Euler 133Liu Hui 45-47Loomis, Elisha 16, 63, 68-70Mathew Stewart 112-113Pappus 107Pierre de Fermat 130-132President Garfield 66-77Pythagoras 27-35Richard Taylor 130Royal Vale Heath 143Sam Lloyd 136Thabit ibn Kurrah 49-52, 111Three greatest 47MeanArithmetic definition 101,103Geometric definition 101,103Harmonic definition 101,103PPappus' TheoremStatement and proof 107-109Used to prove Pythagorean Theorem 110PoetryCompared to prose 13-14Edna St. Vincent Millay 26"Euclid Alone Has Looked… " 26"Euclid's Beauty Revisited"© 87"From Earth to Love" © 3"Love Triangle" © 27"Pearls and Posers"© 136Relation to mathematics text 14"Significance" © 4Power SumsDiscussed 134-135PuzzlesGame Boy Boxel 47Curry's Paradox 168Kurrah's proof as source for 52181
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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
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
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 184 and 185: Puzzles (cont)Pythagorean Magic Squ
Page 186 and 187: Pythagorean Theorem ProofsAlgebraic
Page 188: Trigonometry (cont)Eratosthenes mea
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