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Proof Without Words by Steve Phelps (9)

Proof Without Words by Steve Phelps (9)
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2015-02-20 09:35
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This example demonstrates the Pythagorean Theorem through geometric manipulation by dragging points and a slider. It involves core concepts like the Pythagorean Theorem, difference and slope, ellipses, subtraction, and congruence. The construction utilizes GeoGebra commands such as Point, Intersect, CircleArc, Polygon, Segment, OrthogonalLine, If, Dilate, Vector, and Translate. It is suitable for high school mathematics education to visually understand geometric proofs and dynamically demonstrate mathematical concepts.

GeoGebra Commands & Parameters

  1. Point(xAxis) → A
  2. Intersect(xAxis, yAxis) → B
  3. CircleArc(B, D, A) → c
  4. Point(c) → E
  5. Polygon(A, E, 4) → poly1, a, b, d, e, F, G
  6. Polygon(E, C, 4) → poly2, f, g, h, i, H, I
  7. Polygon(C, A, 4) → poly3, j, k, l, m, J, K
  8. Segment(E, H) → n
  9. Segment(I, C) → p
  10. Intersect(n, p) → L
  11. OrthogonalLine(L, j) → q
  12. OrthogonalLine(L, q) → r
  13. Intersect(r, i) → M
  14. Intersect(q, h) → N
  15. Intersect(r, g) → O
  16. Intersect(q, f) → P
  17. Polygon(M, L, E) → poly4, e_1, m_1, l_1
  18. Polygon(M, I, L) → poly5, l_2, m_2, i_1
  19. Polygon(L, N, I) → poly6, i_2, l_3, n_1
  20. Polygon(L, H, N) → poly7, n_2, l_4, h_1
  21. Polygon(L, O, H) → poly8, h_2, l_5, o
  22. Polygon(L, C, O) → poly9, o_1, l_6, c_1
  23. Polygon(L, P, C) → poly10, c_2, l_7, p_1
  24. Polygon(L, P, E) → poly11, e_2, l_8, p_2
  25. Polygon(F, G, A, E) → poly12, f_1, g_1, a_1, e_3
  26. If(0 ≤ t ≤ 1, 1 - t, If[t > 1, 0, 1]) → d1
  27. Dilate(L, d1, A) → L'
  28. Dilate(L, d1, C) → L'_1
  29. Dilate(L, d1, K) → L'_2
  30. Dilate(L, d1, J) → L'_3
  31. Vector(L, L') → u
  32. Vector(L, L'_3) → v
  33. Vector(L, L'_2) → w
  34. Vector(L, L'_1) → s
  35. Translate(L, u) → L'_4
  36. Translate(P, u) → P'
  37. Translate(C, u) → C'
  38. Polygon(L'_4, P', C') → poly10', c', l'_4, p'
  39. Translate(L, v) → L'_5
  40. Translate(O, v) → O'
  41. Translate(H, v) → H'
  42. Polygon(L'_5, O', H') → poly8', h', l'_5, o'
  43. Translate(L, w) → L'_6
  44. Translate(N, w) → N'
  45. Translate(I, w) → I'
  46. Polygon(L'_6, N', I') → poly6', i', l'_6, n'
  47. Translate(M, s) → M'
  48. Translate(L, s) → L'_7
  49. Translate(E, s) → E'
  50. Polygon(M', L'_7, E') → poly4', e', m', l'_7
  51. If(1 ≤ t ≤ 2, 2 - t, If[t > 2, 0, 1]) → d2
  52. Dilate(L, d2, A) → L'_8
  53. Dilate(L, d2, J) → L'_9
  54. Dilate(L, d2, K) → L'_{10}
  55. Dilate(L, d2, C) → L'_{11}
  56. Vector(L, L'_8) → u_1
  57. Vector(L, L'_9) → v_1
  58. Vector(L, L'_{10}) → w_1
  59. Vector(L, L'_{11}) → b_1
  60. Translate(L, v_1) → L'_{12}
  61. Translate(H, v_1) → H'_1
  62. Translate(N, v_1) → N'_1
  63. Polygon(L'_{12}, H'_1, N'_1) → poly7', n'_1, l'_{12}, h'_1
  64. Translate(L, u_1) → L'_{13}
  65. Translate(C, u_1) → C'_1
  66. Translate(O, u_1) → O'_1
  67. Polygon(L'_{13}, C'_1, O'_1) → poly9', o'_1, l'_{13}, c'_1
  68. Translate(M, w_1) → M'_1
  69. Translate(I, w_1) → I'_1
  70. Translate(L, w_1) → L'_{14}
  71. Polygon(M'_1, I'_1, L'_{14}) → poly5', l'_{14}, m'_1, i'_1
  72. Translate(L, b_1) → L'_{15}
  73. Translate(P, b_1) → P'_1
  74. Translate(E, b_1) → E'_1
  75. Polygon(L'_{15}, P'_1, E'_1) → poly11', e'_1, l'_{15}, p'_1
  76. If(2 ≤ t ≤ 3, 3 - t, If[t > 3, 0, 1]) → d3
  77. Dilate(F, d3, E') → F'
  78. Vector(F, F') → d_1
  79. Translate(F, d_1) → F'_1
  80. Translate(G, d_1) → G'
  81. Translate(A, d_1) → A'
  82. Translate(E, d_1) → E'_2
  83. Polygon(F'_1, G', A', E'_2) → poly12', f'_1, g', a', e'_2

AI Drawing Prompt Reference

Draw a right triangle ABC with point A on the x-axis, point B at the origin, and point C on the y-axis. Construct several polygons using points E, F, G, H, I, and J, adjust the shape with slider t, and use point L to create a series of perpendicular lines and polygons, ultimately forming a series of similar polygons to demonstrate the geometric proof of the Pythagorean theorem.
The prompt above was compiled from the original drawing commands and is for reference only.