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Henry Perigal's Dynamic Proof of Pythagoras' Theorem

Henry Perigal's Dynamic Proof of Pythagoras' Theorem
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2014-07-15 19:57
TriangleCirclePolygonGeometric Construction
In collections GCSE Higher Ch5 Angles & Trigonometry
This example demonstrates a dynamic proof of Pythagoras' theorem created by Henry Perigal, visually splitting and reassembling shapes to illustrate the theorem. Key mathematical concepts include Pythagorean theorem, geometric proof, and shape transformation. GeoGebra commands and objects used in construction are Polygon, Semicircle, Point, OrthogonalLine, Segment, Angle, Line, Intersect, Vector, and Translate. Suitable for high school mathematics education to enhance understanding and application of the Pythagorean theorem.

GeoGebra Commands & Parameters

  1. Polygon(B, A, 4) → mnohoúhelník4, l, m, n, p, H, I
  2. Semicircle(A, B) → c
  3. Point(c) → C
  4. Polygon(A, B, C) → mnohoúhelník1, c_1, a_1, b
  5. OrthogonalLine(C, c_1) → d_1
  6. Segment(A, B) → a
  7. Angle(A, C, B) → α
  8. Polygon(C, B, 4) → mnohoúhelník2, d, e, f, g, D, E
  9. Polygon(A, C, 4) → mnohoúhelník3, h, i, j, k, F, G
  10. Line(D, E) → q
  11. Line(G, F) → r
  12. Segment(E, B) → s
  13. Segment(D, C) → t
  14. Intersect(s, t) → J
  15. Line(J, n) → b_1
  16. OrthogonalLine(J, b_1) → e_1
  17. Intersect(f, e_1) → K
  18. Intersect(b_1, e) → L
  19. Intersect(b_1, g) → M
  20. Intersect(e_1, a_1) → N
  21. Segment(K, N) → f_1
  22. Segment(M, L) → g_1
  23. Polygon(C, N, J, M) → mnohoúhelník5, c_2, n_1, j_1, m_1
  24. Polygon(K, J, L, D) → mnohoúhelník6, k_1, j_2, l_1, d_2
  25. Polygon(K, E, M, J) → mnohoúhelník7, k_2, e_2, m_2, j_3
  26. Polygon(B, N, J, L) → mnohoúhelník8, b_2, n_2, j_4, l_2
  27. Segment(F, A) → h_1
  28. Segment(G, C) → i_1
  29. Intersect(h_1, i_1) → O
  30. Segment(I, A) → p_1
  31. Segment(H, B) → q_1
  32. Intersect(p_1, q_1) → P
  33. Vector(O, P) → u
  34. Translate(mnohoúhelník3, u) → mnohoúhelník3'
  35. Translate(mnohoúhelník3, Vector[(u * (o - 1.2))]) → mnohoúhelník3'_1
  36. Vector(J, B) → v
  37. Translate(C, v) → C'
  38. Translate(N, v) → N'
  39. Translate(J, v) → J'
  40. Translate(M, v) → M'
  41. Polygon(C', N', J', M') → mnohoúhelník5', c', n', j', m'
  42. Translate(C, Vector[(v * (o - 0.2))]) → C'_1
  43. Translate(N, Vector[(v * (o - 0.2))]) → N'_1
  44. Translate(J, Vector[(v * (o - 0.2))]) → J'_1
  45. Translate(M, Vector[(v * (o - 0.2))]) → M'_1
  46. Polygon(C'_1, N'_1, J'_1, M'_1) → mnohoúhelník5'_1, c'_1, n'_1, j'_1, m'_1
  47. Vector(J, H) → w
  48. Translate(K, Vector[(w * (o - 0.2))]) → K'
  49. Translate(J, Vector[(w * (o - 0.2))]) → J'_2
  50. Translate(L, Vector[(w * (o - 0.2))]) → L'
  51. Translate(D, Vector[(w * (o - 0.2))]) → D'
  52. Polygon(K', J'_2, L', D') → mnohoúhelník6', k', j'_2, l', d'
  53. Translate(K, w) → K'_1
  54. Translate(J, w) → J'_3
  55. Translate(L, w) → L'_1
  56. Translate(D, w) → D'_1
  57. Polygon(K'_1, J'_3, L'_1, D'_1) → mnohoúhelník6'_1, k'_1, j'_3, l'_1, d'_1
  58. Vector(J, I) → z
  59. Translate(K, z) → K'_2
  60. Translate(E, z) → E'
  61. Translate(M, z) → M'_2
  62. Translate(J, z) → J'_4
  63. Polygon(K'_2, E', M'_2, J'_4) → mnohoúhelník7', k'_2, e', m'_2, j'_4
  64. Translate(K, Vector[(z * (o - 0.2))]) → K'_3
  65. Translate(E, Vector[(z * (o - 0.2))]) → E'_1
  66. Translate(M, Vector[(z * (o - 0.2))]) → M'_3
  67. Translate(J, Vector[(z * (o - 0.2))]) → J'_5
  68. Polygon(K'_3, E'_1, M'_3, J'_5) → mnohoúhelník7'_1, k'_3, e'_1, m'_3, j'_5
  69. Vector(J, A) → u_1
  70. Translate(B, u_1) → B'
  71. Translate(N, u_1) → N'_2
  72. Translate(J, u_1) → J'_6
  73. Translate(L, u_1) → L'_2
  74. Polygon(B', N'_2, J'_6, L'_2) → mnohoúhelník8', b', n'_2, j'_6, l'_2
  75. Translate(B, Vector[(u_1 * (o - 0.2))]) → B'_1
  76. Translate(N, Vector[(u_1 * (o - 0.2))]) → N'_3
  77. Translate(J, Vector[(u_1 * (o - 0.2))]) → J'_7
  78. Translate(L, Vector[(u_1 * (o - 0.2))]) → L'_3
  79. Polygon(B'_1, N'_3, J'_7, L'_3) → mnohoúhelník8'_1, b'_1, n'_3, j'_7, l'_3

AI Drawing Prompt Reference

Draw a dynamic proof of the Pythagorean theorem created by Henry Perigal, visually splitting and reassembling shapes to illustrate the theorem. The diagram includes a quadrilateral and two semicircles, with triangles and other geometric shapes derived from them. Draw the quadrilateral and semicircles first, then add perpendicular lines, edges, and angles, and find their intersections. Finally, show how the shapes can be reassembled by translating and rotating them to prove the Pythagorean theorem.
The prompt above was compiled from the original drawing commands and is for reference only.