The Pythagorean Theorem - Educational Outreach

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Letxcp(cpbe the x component of an arbitrary critical pointxcp, y ) . Now xcpmust be such that x cp Cx [ C x ] cp0 0 inorder for the quantity 0 . This in turn allowstwo real-number values foronly thosecpcpy values where y 0ycp. In light of Figure 2.33,cpare of interest.(x cp, C) & G(C) < 0G(y) isdefined onthis line(x cp, y cp)&G(y cp) = 0(x cp, 0) & G(0) > 0Figure 2.36: Behavior of G on IntDDefine the continuous quadratic function2 x [ C x ] on the vertical line segmentG( y)ycpconnecting the two points ,0)cp( cpx and ( x cp, C)on BndD asshown in Figure 2.36. On the lower segment, we haveG ( 0) xcp[C xcp] 0 for all xcpin 0 x cp C . On theupper segment, we have that G(C) x [ C x ] C2 0for allx in x cp Ccp0 . This is since the maximum thatG( x,C)can achieve for any xcpin 0 x cp C is 3C2 / 4per the optimization techniques of single-variable calculus.Now, by the intermediate value theorem, there must be avalue0 y* C where G(y*) x [ C x ] ( y*)2 076cpcpcp cp.By inspection, the associated point ( x cp, y*)is in IntD ,and, thus, by definition is part of the locus of critical pointswith x , y*) ( x , y ) .(cpcp cp
Sincexcpwas chosen on an arbitrary basis, all criticalpoints , y )in IntD .3x defined by x [ C x ] y 0(cp cpcpcp2 cp : Thus far, we have used the techniques of differentialcalculus (both single and multi-variable) to establish alocus of critical points x [ C x ] y2 0 lying entirelycpwithin IntD . What is the relationship of each pointx , y ) to the <strong>Pythagorean</strong> <strong>The</strong>orem? As observed at the(cp cpend of Step 1, one can immediately state the following:cpcplieF(xxAAmin22min, y22min[ C x B Bmin C) 0 ] y2 C22min 0 0 .M = (x cp, y cp)ηθγ = θ + ηLy cpα βx cpP C - x cpNFigure 2.37: D and Locus of Critical Points77
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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
Page 19 and 20:
1) Consider the Squares“If it was
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s090045 0ss45450450090Figure 1.3: O
Page 23 and 24:
Notice that the right-triangle prop
Page 25 and 26:
The middle square (minus the donut
Page 27 and 28: Our proof in Chapter 1 has been by
Page 29 and 30: 2) Four Thousand Years of Discovery
Page 31 and 32: The proof Pythagoras is thought to
Page 33 and 34: cabcFigure 2.4: Annotated Square wi
Page 35 and 36: Figure 2.6 is the diagram for a sec
Page 37 and 38: 1x:b2:Area(CBA)3:Area( ABC)Area(ABC
Page 39 and 40: Euclid’s proof of the Pythagorean
Page 41 and 42: First we establish that the two tri
Page 43 and 44: 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: Thus, the critical point ( C 2 ,0)i
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 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
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a2Law of Cosines c2 b2 2bccos()The
Page 133 and 134:
An example of a Diophantine equatio
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However, Euler’s Conjecture did n
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4) Pearls of Fun and WonderPearls o
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We have achieved five-digit accurac
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140212213116)74370116)(74370(370)74
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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
Page 150 and 151:
Figure 4.9 depicts Eratosthenes’
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---High above the earth, we see the
Page 154 and 155:
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
Page 162 and 163:
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
Page 172 and 173:
A] Greek AlphabetGREEK LETTERUpper
Page 174 and 175:
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 &
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 186 and 187:
Pythagorean Theorem ProofsAlgebraic
Page 188:
Trigonometry (cont)Eratosthenes mea
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The Pythagorean Theorem - Educational Outreach

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