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Dynamic Illustration of a Chord & Tangent Theorem

Dynamic Illustration of a Chord & Tangent Theorem
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2016-10-23 11:43
CircleAngleTrigonometryGeometric Construction
This example dynamically illustrates a less commonly discussed theorem that relates both chords and tangents. The math concepts involved include circle, angle, congruence, rotation, similarity, and more. It is constructed using GeoGebra commands and objects such as Circle, Point, Tangent, Segment, Angle, Rotate, CircleArc, Intersect, CircleSector, Dilate, Ray, and Distance. Suitable for high school and above students to visually understand complex geometric theorems and their proofs.

GeoGebra Commands & Parameters

  1. Circle(A_1, R) → c
  2. Point(c) → B
  3. Point(c) → A
  4. Tangent(A, c) → j,
  5. Segment(A, B) → h
  6. Angle(B, A_1, A) → α
  7. Rotate(B, α / 2, A_1) → D'
  8. CircleArc(A_1, A, B) → e
  9. Point(e) → N
  10. Tangent(N, c) → g,
  11. Intersect(g, j) → F
  12. Segment(D', F) → k
  13. Intersect(h, k) → P
  14. Circle(P, A) → p_2
  15. Intersect(p_2, h) → V, W
  16. Segment(A_1, R) → f
  17. CircleArc(A_1, B, A) → d
  18. Angle(B, A_1, D') → β
  19. Tangent(B, c) → i,
  20. Intersect(g, i) → G
  21. Segment(D', G) → l
  22. Intersect(h, l) → Q
  23. Segment(A, P) → m
  24. Segment(P, Q) → n
  25. Segment(Q, B) → p
  26. Segment(A_1, B) → q
  27. Segment(A_1, D') → r
  28. Segment(A_1, A) → s
  29. Circle(A_1, (0.2 * r)) → t
  30. Intersect(t, q, 1) → J
  31. Intersect(t, r, 1) → K
  32. CircleSector(A_1, J, K) → c_1
  33. Dilate(A_1, If[0 < b ≤ 1, 1 - b, 0], A) → A'
  34. Dilate(A_1, If[0 < b ≤ 1, 1 - b, 0], B) → J'
  35. Segment(A_1, J') → f_1
  36. Segment(A_1, A') → g_1
  37. Dilate(A_1, If[6 < b ≤ 7, 7 - b, 0], D') → A'_1
  38. Segment(A_1, A'_1) → h_1
  39. Circle(B, (0.2 * r)) → e_1
  40. Circle(A, (0.2 * r)) → k_1
  41. Intersect(k_1, j) → L, M
  42. Intersect(k_1, g_1) → N_1,
  43. Intersect(e_1, i) → O, P_1
  44. Intersect(e_1, f_1) → Q_1,
  45. CircleSector(A, N_1, M) → p_1
  46. CircleSector(B, O, Q_1) → q_1
  47. Rotate(q_1, If[1.5 < b ≤ 2.5, ((b - 1.5) * π / 2), π / 2], B) → s_1
  48. Dilate(A_1, If[3.5 < b ≤ 4.5, 4.5 - b, 0], N) → A'_2
  49. Segment(A_1, A'_2) → i_1
  50. Circle(N, (0.2 * r)) → t_1
  51. Intersect(t_1, g, 1) → R_1
  52. Intersect(t_1, i_1, 1) → S
  53. CircleSector(N, R_1, S) → c_2
  54. Dilate(N, If[2.5 < b ≤ 3.5, 3.5 - b, 0], G) → C'
  55. Dilate(N, If[2.5 < b ≤ 3.5, 3.5 - b, 0], F) → C'_1
  56. Segment(N, C') → j_1
  57. Segment(N, C'_1) → l_1
  58. Rotate(c_2, If[5 < b ≤ 6, ((b - 5) * π / 2), π / 2], N) → d_2
  59. Rotate(c_1, If[7.5 < b ≤ 8.5, ((b - 7.5) * β), β], A_1) → d_1
  60. Ray(B, O) → m_1
  61. Circle(G, 1) → e_2
  62. Intersect(e_2, i, 2) → T
  63. Circle(F, 1) → f_2
  64. Intersect(f_2, j, 1) → U
  65. Ray(A, M) → n_1
  66. Ray(F, U) → a_1
  67. Ray(G, T) → b_1
  68. Segment(B, G) → g_2
  69. Segment(A, F) → h_2
  70. Rotate(p_1, If[1.5 < b ≤ 2.5, ((-(b - 1.5)) * π / 2), ((-π)) / 2], A) → r_1
  71. Dilate(F, If[8.5 < b ≤ 9.5, 9.5 - b, 0], D') → D''
  72. Segment(D', D'') → i_2
  73. Dilate(G, If[9.5 < b ≤ 10.5, 10.5 - b, 0], D') → D''_1
  74. Segment(D', D''_1) → j_2
  75. Distance(D', P) → o
  76. Distance(D', F) → u
  77. Distance(D', Q) → w
  78. Distance(D', G) → o_1
  79. Dilate(A, If[10.5 < b ≤ 11.5, 11.5 - b, 0], P) → E'
  80. Segment(A, E') → k_2
  81. Dilate(P, If[11.5 < b ≤ 12.5, 12.5 - b, 0], W) → H'
  82. Segment(P, H') → l_2
  83. Dilate(Q, If[13.5 < b ≤ 14.5, 14.5 - b, 0], B) → I'
  84. Segment(Q, I') → m_2
  85. Dilate(W, If[12.5 < b ≤ 13.5, 13.5 - b, 0], Q) → W'
  86. Segment(W, W') → n_2
  87. Rotate(A, If[15 < b ≤ 16, ((b - 15) * π), π], P) → E'_1
  88. Rotate(E', If[15 < b ≤ 16, ((b - 15) * π), π], P) → E''
  89. Segment(E'_1, E'') → q_2
  90. Rotate(Q, If[16 < b ≤ 17, ((-(b - 16)) * π), (-π)], Q) → I'_1
  91. Rotate(I', If[16 < b ≤ 17, ((-(b - 16)) * π), (-π)], Q) → I''
  92. Segment(I'_1, I'') → r_2
  93. Segment(F, D'') → s_2
  94. Segment(G, D''_1) → t_2