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Thales' Theorem Dynamic Demonstration + Sequel

Thales' Theorem Dynamic Demonstration + Sequel
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2017-04-15 08:52
TriangleCircleAngleGeometric Construction
In collections Circles (Theorems)
This example dynamically demonstrates Thales' theorem problem, showcasing geometric relationships through interactive illustrations. It involves core math concepts like circles, triangles, angles, rotation, and transformation. Construction highlights include commands such as Circle, Point, Line, Intersect, Rotate, and Translate. Suitable for high school mathematics education to visually understand geometric theorems and transformations.

GeoGebra Commands & Parameters

  1. Circle(A, B) → c
  2. Point(c) → C
  3. Line(C, A) → a
  4. Intersect(c, a, 2) → D
  5. CircleArc(A, C, D) → d
  6. CircleArc(A, D, C) → e
  7. Point(e) → E
  8. Segment(C, D) → h
  9. Dilate(C, If[0 < b ≤ 1, 1 - b, 0], A) → C'
  10. Segment(C, C') → i
  11. Dilate(A, If[1 < b ≤ 2, 2 - b, 0], D) → A'
  12. Segment(A, A') → j
  13. Rotate(C, If[2 < b ≤ 3, ((-(b - 2)) * π), (-π)], A) → C'_1
  14. Rotate(C', If[2 < b ≤ 3, ((-(b - 2)) * π), (-π)], A) → C''
  15. Segment(C'_1, C'') → i'
  16. Rotate(C, If[3 < b ≤ 4, ((b - 3) * π), π], A) → C'_2
  17. CircleArc(A, C, C'_2) → k
  18. Dilate(D, If[5 < b ≤ 6, 6 - b, 0], E) → D'
  19. Segment(D, D') → f
  20. Ray(D, E) → g
  21. Point(g) → F
  22. Ray(E, F) → l
  23. Dilate(C, If[5 < b ≤ 6, 6 - b, 0], E) → C'_3
  24. Segment(C, C'_3) → m
  25. Min({m, f}) → n
  26. Circle(E, (0.2 * n)) → p
  27. Intersect(p, l) → G,
  28. Intersect(c, m, 1) → H
  29. Intersect(p, m) → I,
  30. CircleSector(E, G, I) → q
  31. Rotate(q, If[6.5 < b ≤ 7.5, ((b - 6.5) * π / 2), π / 2], E) → q'
  32. Point(cc) → J
  33. Line(C, D) → r
  34. OrthogonalLine(J, r) → s
  35. Intersect(s, r) → K
  36. Line(C, E) → t
  37. OrthogonalLine(J, t) → f_1
  38. Intersect(t, f_1) → L
  39. Line(K, L) → g_1
  40. Line(J, E) → h_1
  41. Intersect(g_1, h_1) → M
  42. Angle(A, M, J) → α
  43. Line(A, M) → i_1
  44. Circle(C, 1) → c_1
  45. Circle(E, 1) → d_1
  46. Intersect(d_1, t, 2) → N
  47. Intersect(c_1, t, 1) → O
  48. Ray(E, N) → j_1
  49. Ray(C, O) → k_1
  50. Dilate(J, If[8 < b ≤ 9, 9 - b, 0], K) → J'
  51. Segment(J, J') → l_1
  52. Dilate(E, If[9 < b ≤ 10, 10 - b, 0], K) → E'
  53. Vector(E, E') → u
  54. Translate(q', u) → q''
  55. Line(E, r) → m_1
  56. Point(m_1) → P
  57. Angle(D, E, P) → β
  58. Rotate(q'', If[10 < b ≤ 11, ((b - 10) * β), β], K) → q'''
  59. Mirror(J, r) → J'_1
  60. Dilate(K, If[9 < b ≤ 10, 10 - b, 0], J'_1) → K'
  61. Segment(K, K') → n_1
  62. Dilate(J, If[11 < b ≤ 12, 12 - b, 0], L) → J'_2
  63. Segment(J, J'_2) → p_1
  64. Dilate(E, If[12 < b ≤ 13, 13 - b, 0], L) → E'_1
  65. Vector(E, E'_1) → v
  66. Translate(q', v) → q''_1
  67. Dilate(J, If[13 < b ≤ 14, 14 - b, 0], E) → J'_3
  68. Segment(J, J'_3) → q_1
  69. Dilate(K, If[14 < b ≤ 15, 15 - b, 0], M) → K'_1
  70. Segment(K, K'_1) → r_1
  71. Dilate(M, If[15 < b ≤ 16, 16 - b, 0], L) → M'
  72. Segment(M, M') → s_1
  73. Circle(A, 1) → e_1
  74. Circle(M, 1) → t_1
  75. Intersect(t_1, c, 1) → Q
  76. Intersect(t_1, i_1) → R, S
  77. Intersect(e_1, i_1) → T, U
  78. Dilate(A, If[16 < b ≤ 17, 17 - b, 0], M) → A'_1
  79. Segment(A, A'_1) → a_1
  80. Ray(M, S) → b_1
  81. Ray(A, T) → f_2
  82. Dilate(E, If[17 < b ≤ 18, 18 - b, 0], M) → E'_2
  83. Vector(E, E'_2) → w
  84. Translate(q', w) → q''_2
  85. Line(E, f_2) → g_2
  86. Point(g_2) → V
  87. Angle(V, E, L) → δ
  88. Rotate(q''_2, If[18 < b ≤ 19, ((-(b - 18)) * δ), (-δ)], M) → c_2
  89. Polygon(C, D, E) → poly1, e_2, c_3, d_2
  90. CircleArc(A, E, C) → h_2
  91. CircleArc(A, D, E) → k_2
  92. CircleArc(A, C, D) → p_2
  93. Polygon(C, D, E) → poly2, e_3, c_4, d_3
  94. Circle(C, 1) → q_2
  95. Circle(D, 1) → r_2
  96. Intersect(r_2, a) → A_1, B_1
  97. Ray(D, B_1) → i_2