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非传统弦与切线定理动态演示

非传统弦与切线定理动态演示
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2016-10-23 11:43
圆角三角函数几何作图
所属专题 Circles (Theorems)
①本范例动态演示了一个不常被提及的、同时关联弦与切线的定理。②涉及的核心数学概念包括圆、角、共线、旋转、相似等。③作图要点：运用了Circle（圆）、Point（点）、Tangent（切线）、Segment（线段）、Angle（角）、Rotate（旋转）、CircleArc（圆弧）、Intersect（交点）、CircleSector（扇形）、Dilate（缩放）、Ray（射线）、Distance（距离）等GeoGebra命令与对象。④适合高中及以上数学学习者，用于理解复杂几何定理的直观表现与证明思路。

GeoGebra 命令及参数

  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