The Pythagorean Theorem - Educational Outreach

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Figures…continuedNumber and TitlePage2.1: Egyptian Knotted Rope, Circa 2000 BCE 272.2: The First Proof by Pythagoras 292.3: An Alternate Visual Proof by Pythagoras 302.4: Annotated Square within a Square 312.5: Algebraic Form of the First Proof 322.6: A Rectangular Dissection Proof 332.7: Twin Triangle Proof 342.8: Euclid’s Windmill without Annotation 362.9 Pondering Squares and Rectangles 372.10: Annotated Windmill 382.11: Windmill Light 402.12: Euclid’s Converse Theorem 432.13: Liu Hui’s Diagram with Template 452.14: Packing Two Squares into One 462.15: The Stomachion Created by Archimedes 482.16: Kurrah Creates the Brides Chair 492.17: Packing the Bride’s Chair into the Big Chair 502.18: Kurrah’s Operation Transformation 512.19: The Devil’s Teeth 522.20: Truth versus Legend 532.21: Bhaskara’s Real Power 542.22: Leonardo da Vinci’s Symmetry Diagram 552.23: Da Vinci’s Proof in Sequence 562.24: Subtle Rotational Symmetry 572.25: Legendre’s Diagram 582.26: Barry Sutton’s Diagram 592.27: Diagram on Henry Perigal’s Tombstone 612.28: Annotated Perigal Diagram 622.29: An Example of Pythagorean Tiling 642.30: Four Arbitrary Placements… 642.31: Exposing Henry’s Quadrilaterals 652.32: President Garfield’s Trapezoid 662.33: Carolyn’s Cauliflower 712.34: Domain D of F 728
Figures…continuedNumber and TitlePage2.35: Carolyn’s Cauliflower for y = 0 752.36: Behavior of G on IntD 762.37: Domain and Locus of Critical Points 772.38: Logically Equivalent Starting Points 792.39: Walking from Inequality to Equality… 802.40: Shearing a Rectangle 832.41: A Four-Step Shearing Proof 843.1: Three Squares, Three Crosses 883.2: Pythagorean Triples 903.3 Inscribed Circle Theorem 963.4 Three Dimensional Pythagorean Theorem 983.5: Three-Dimensional Distance Formula 1003.6: The Three Pythagorean Means 1013.7: The Two Basic Triangle Formulas 1043.8: Schematic of Hero’s Steam Engine 1043.9: Diagram for Heron’s Theorem 1053.10: Diagram for Pappus’ Theorem 1083.11: Pappus Triple Shear-Line Proof 1093.12: Pappus Meets Pythagoras 1103.13: Diagram for Kurrah’s Theorem 1113.14: Diagram for Stewart’s Theorem 1133.15: Trigonometry via Unit Circle 1183.16: Trigonometry via General Right Triangle 1203.17: The Cosine of the Sum 1223.18: An Intricate Trigonometric Decomposition 1253.19: Setup for the Law of Sines and Cosines 1274.1: Triangular Lake and Solution 1374.2: Pure and Perfect 4x4 Magic Square 1414.3: 4x4 Magic Patterns 1424.4: Pythagorean Magic Squares 1434.5: The Schoolhouse Flagpole 1444.6: Off-Limits Windmill 1454.7 Across the Thorns and Nettles 1464.8: Eratosthenes’s Egypt 1479
Page 1: The PythagoreanTheoremCrown Jewel o
Page 4: The Pythagorean TheoremCrown Jewel
Page 7 and 8: Table of ContentsList of Tables and
Page 9: List of Tables and FiguresTablesNum
Page 13 and 14: List of Proofs and DevelopmentsSect
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
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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
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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
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Figure 4.9 depicts Eratosthenes’
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---High above the earth, we see the
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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
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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
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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 &
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Recreational Mathematics11. Pickove
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GGolden RatioDefined and algebraica
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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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