The Pythagorean Theorem - Educational Outreach

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Figures…continuedNumber and TitlePage4.9: Eratosthenes Measures the Earth 1484.10: View Distance to Earth’s Horizon 1494.11: Measuring the Moon 1504.12: From Moon to Sun 1534.13: From Sun to Alpha Centauri 1554.14: The Golden Ratio 1574.15: Two Golden Triangles 1584.16: Triangular Phi 1604.17: Pythagorean PI 1614.18: Recursive Hypotenuses 1624.19: Pythagorean Spiral 165E.1: Beauty in Order 166E.2: Curry’s Paradox 167A.0: The Tangram 16910
List of Proofs and DevelopmentsSection and TopicPage1.1: Speculative Genesis of Pythagorean Theorem 232.1: Primary Proof by Pythagoras* 292.1: Alternate Proof by Pythagoras* 302.1: Algebraic Form of Primary Proof* 322.1: Algebraic Rectangular Dissection Proof* 332.1: Algebraic Twin-Triangle Proof* 342.2: Euclid’s Windmill Proof* 392.2: Algebraic Windmill Proof* 402.2: Euclid’s Proof of the Pythagorean Converse 422.3: Liu Hui’s Packing Proof* 452.3: Stomachion Attributed to Archimedes 482.4: Kurrah’s Bride’s Chair 492.4: Kurrah’s Transformation Proof* 502.5: Bhaskara’s Minimal Algebraic Proof* 542.6: Leonardo da Vinci’s Skewed-Symmetry Proof * 562.7: Legendre’s Embedded-Similarity Proof* 582.7: Barry Sutton’s Radial-Similarity Proof* 592.8: Henry Perigal’s Quadrilateral-Dissection Proof*622.8: Two Proofs by Pythagorean Tiling* 642.9: President Garfield’s Trapezoid Proof* 662.10: Cauliflower Proof using Calculus* 712.11: Four-Step Shearing Proof* 843.1: Pythagorean Extension to Similar Areas 883.2: Formula Verification for Pythagorean Triples 913.3: Proof of the Inscribed Circle Theorem 963.4: Three-Dimension Pythagorean Theorem 983.4: Formulas for Pythagorean Quartets 993.4: Three-Dimensional Distance Formula 1003.5: Geometric Development of the Three Means 1013.6: Proof of Heron’s Theorem 1063.6: Proof of Pappus’ General Triangle Theorem 1083.6: Proof of Pythagorean Theorem 110Using Pappus’ Theorem*11
Page 1: The PythagoreanTheoremCrown Jewel o
Page 4: The Pythagorean TheoremCrown Jewel
Page 7 and 8: Table of ContentsList of Tables and
Page 9 and 10: List of Tables and FiguresTablesNum
Page 11: Figures…continuedNumber and Title
Page 15 and 16: PrefaceThe Pythagorean Theorem has
Page 17 and 18: 1 : 2x2x2****22 13x 7 13x 7 0 : a
Page 19 and 20: 1) Consider the Squares“If it was
Page 21 and 22: 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
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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
Page 103 and 104:
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
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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
Page 158 and 159:
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
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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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