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Perigal Pythagorean Using Dilations And Vectors

Perigal Pythagorean Using Dilations And Vectors
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2016-11-30 02:22
VectorTransformationGeometric Construction
This example dynamically demonstrates the Perigal Pythagorean Theorem, revealing the geometric intuition behind the area relationships of three squares. It involves core mathematical concepts such as dilation transformations, vector operations, special point construction, and coordinate geometry. The construction utilizes GeoGebra commands and objects like segments, semicircles, points, polygons, midpoints, lines, intersections, dilations, vectors, and translations. Suitable for high school and above learners to understand the application of transformations and vectors in geometric proofs.

GeoGebra Commands & Parameters

  1. Segment(A, B) → f
  2. Semicircle(A, B) → c
  3. Point(c) → C
  4. Segment(A, C) → g
  5. Segment(C, B) → h
  6. Polygon(A, C, 4) → poly1, i, j, k, l, D, E
  7. Polygon(C, B, 4) → poly2, m, n, p, q, F, G
  8. Polygon(B, A, 4) → poly3, r, s, t, a, H, I
  9. Segment(G, B) → b
  10. Midpoint(b) → J
  11. Line(J, a) → d
  12. Line(J, f) → e
  13. Intersect(e, q) → K
  14. Intersect(d, p) → L
  15. Intersect(n, e) → M
  16. Intersect(d, h) → N
  17. Polygon(C, K, J, N) → poly4, c_1, k_1, j_1, n_1
  18. Polygon(N, J, M, B) → poly5, n_2, j_2, m_1, b_1
  19. Polygon(F, M, J, L) → poly6, f_1, m_2, j_3, l_1
  20. Polygon(G, L, J, K) → poly7, g_1, l_2, j_4, k_2
  21. Segment(J, A) → h_1
  22. Segment(J, B) → i_1
  23. Segment(J, H) → p_1
  24. Segment(J, I) → q_1
  25. Dilate(A, o, J) → A'
  26. Dilate(B, o, J) → B'
  27. Dilate(H, o, J) → H'
  28. Dilate(I, o, J) → I'
  29. Vector(J, A') → u
  30. Vector(J, B') → v
  31. Vector(J, I') → w
  32. Vector(J, H') → u_1
  33. Translate(C, v_1) → C'
  34. Translate(K, v_1) → K'
  35. Translate(J, v_1) → J'
  36. Translate(N, v_1) → N'
  37. Polygon(C', K', J', N') → poly4', c', k', j', n'
  38. Translate(C', v) → C''
  39. Translate(K', v) → K''
  40. Translate(J', v) → J''
  41. Translate(N', v) → N''
  42. Polygon(C'', K'', J'', N'') → poly4'', c'', k'', j'', n''
  43. Translate(N, u) → N'_1
  44. Translate(J, u) → J'_1
  45. Translate(M, u) → M'
  46. Translate(B, u) → B'_1
  47. Polygon(N'_1, J'_1, M', B'_1) → poly5', n'_1, j'_1, m', b'_1
  48. Translate(G, w) → G'
  49. Translate(L, w) → L'
  50. Translate(J, w) → J'_2
  51. Translate(K, w) → K'_1
  52. Polygon(G', L', J'_2, K'_1) → poly7', g', l', j'_2, k'_1
  53. Translate(F, u_1) → F'
  54. Translate(M, u_1) → M'_1
  55. Translate(J, u_1) → J'_3
  56. Translate(L, u_1) → L'_1
  57. Polygon(F', M'_1, J'_3, L'_1) → poly6', f', m'_1, j'_3, l'_1
  58. Segment(A, I) → r_1
  59. Midpoint(r_1) → O
  60. Segment(D, A) → s_1
  61. Midpoint(s_1) → P
  62. Segment(P, O) → t_1
  63. Dilate(O, o, P) → O'
  64. Vector(P, O') → w_1
  65. Translate(poly1, w_1) → poly1'
  66. Segment(A, H) → a_1
  67. Segment(H, I) → d_1
  68. Segment(I, B) → e_1