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动态圆内切线与外切线角度关系演示

动态圆内切线与外切线角度关系演示
几何画板及绘图内容加载中，请稍等……
2017-06-27 07:33
圆角向量几何作图
①本范例动态演示了在圆内切线与外切线情境下，两个角之间的变化关系；②涉及核心数学概念包括圆、切线、角、向量、几何变换等；③通过 GeoGebra 命令如 Circle, Segment, PointIn, Distance, Tangent, Intersect, Dilate, Angle, Line, Rotate 等实现图形构建与动态调整；④适合高中及以上数学学习者，用于理解几何关系与动态几何作图。

GeoGebra 命令及参数

  1. Circle(A, B) → c
  2. Segment(A, B) → f
  3. PointIn(a) → C
  4. Distance(C, c) → b
  5. Circle(C, d) → e
  6. Tangent(c, e) → g, h, i, j
  7. Intersect(c, h) → D
  8. Intersect(c, j) → E, N
  9. Intersect(e, j) → F
  10. Intersect(e, h) → G, L
  11. Intersect(e, g) → H, K
  12. Intersect(e, i) → I
  13. Intersect(c, i) → J, M
  14. Intersect(c, g) → O
  15. Circle(D, 2) → p
  16. Circle(E, 2) → q
  17. Circle(O, 2) → r
  18. Circle(K, 2) → s
  19. Circle(I, 2) → t
  20. Circle(F, 2) → c_1
  21. Circle(G, 2) → d_1
  22. Circle(J, 2) → e_1
  23. Intersect(e_1, i, 1) → P
  24. Intersect(e_1, i, 2) → Q
  25. Intersect(p, h, 1) → R
  26. Intersect(p, h, 2) → S
  27. Intersect(c_1, j, 2) → T
  28. Intersect(c_1, j, 1) → U
  29. Intersect(d_1, h, 1) → V
  30. Intersect(d_1, h, 2) → W
  31. Intersect(s, g, 2) → Z
  32. Intersect(s, g, 1) → A_1
  33. Intersect(r, g, 2) → B_1
  34. Intersect(r, g, 1) → C_1
  35. Intersect(q, j, 2) → D_1
  36. Intersect(q, j, 1) → E_1
  37. Dilate(A, If[0 < k ≤ 1, 1 - k, 0], D) → A'
  38. Dilate(A, If[0 < k ≤ 1, 1 - k, 0], M) → A'_1
  39. Dilate(A, If[0 < k ≤ 1, 1 - k, 0], E) → A'_2
  40. Dilate(A, If[0 < k ≤ 1, 1 - k, 0], O) → A'_3
  41. Angle(A, O, C_1) → α
  42. Angle(A, E, E_1) → β
  43. Angle(A, D, S) → δ
  44. Intersect(t, i, 1) → F_1
  45. Angle(C, I, F_1) → ζ
  46. Angle(C, G, W) → θ_1
  47. Dilate(C, If[0 < k ≤ 1, 1 - k, 0], G) → C'
  48. Dilate(C, If[0 < k ≤ 1, 1 - k, 0], F) → C'_1
  49. Dilate(C, If[0 < k ≤ 1, 1 - k, 0], I) → C'_2
  50. Dilate(C, If[0 < k ≤ 1, 1 - k, 0], K) → C'_3
  51. Intersect(i, j) → G_1
  52. Intersect(i, j) → H_1
  53. Line(C, H) → k_1
  54. Line(O, E) → l_1
  55. Intersect(e, ee) → J_1, , ,
  56. Intersect(e, l_1) → K_1, L_1
  57. Segment(H, K_1) → l
  58. Segment(C, L_1) → m
  59. Min(m, l) → n
  60. Circle(H, (0.3 * n)) → f_1
  61. Segment(C, F) → g_1
  62. Intersect(f_1, g) → I_1, M_1
  63. Intersect(f_1, l) → N_1,
  64. Line(F, C) → i_1
  65. Intersect(e, i_1) → O_1, P_1
  66. Angle(N_1, H, I_1) → ε
  67. Rotate(P_1, ε, C) → Q_1
  68. Angle(P_1, C, Q_1) → γ
  69. Dilate(F, If[0 < k ≤ 1, 1 - k, 0], C) → F'
  70. Segment(F, F') → h_1
  71. Dilate(C, If[1 < k ≤ 2, 2 - k, 0], P_1) → C'_4
  72. Segment(C, C'_4) → j_1
  73. Dilate(O, If[2 < k ≤ 3, 3 - k, 0], E) → O'
  74. Segment(O, O') → m_1
  75. Dilate(E, If[3 < k ≤ 4, 4 - k, 0], K_1) → E'
  76. Segment(E, E') → n_1
  77. Dilate(K_1, If[4 < k ≤ 5, 5 - k, 0], L_1) → R_1
  78. Segment(K_1, R_1) → p_1
  79. Dilate(H, If[5 < k ≤ 6, 6 - k, 0], K_1) → H'
  80. Segment(H, H') → q_1
  81. Line(O, K_1) → s_1
  82. Angle(P_1, C, L_1) → η
  83. Rotate(I_1, If[6 < k ≤ 7, ((-(k - 6)) * ε), (-ε)], H) → S_1
  84. CircleSector(H, S_1, I_1) → r_1
  85. Dilate(C, If[7 < k ≤ 8, 8 - k, 0], L_1) → C'_5
  86. Segment(C, C'_5) → t_1
  87. Dilate(H, If[8 < k ≤ 9, 9 - k, 0], C) → H'_1
  88. Vector(H, H'_1) → w
  89. Translate(r_1, w) → c_2
  90. Line(H, j_1) → a_1
  91. Line(H, t_1) → b_1
  92. Point(b_1) → T_1
  93. Angle(I_1, H, T_1) → θ
  94. Rotate(c_2, If[9 < k ≤ 10, ((k - 9) * θ), θ], C) → d_2
  95. Dilate(C, If[10 < k ≤ 11, 11 - k, 0], Q_1) → C'_6
  96. Segment(C, C'_6) → f_2
  97. Dilate(H, If[11 < k ≤ 12, 12 - k, 0], C) → H'_2
  98. Vector(H, H'_2) → u_1
  99. Translate(r_1, u_1) → e_2
  100. Rotate(e_2, If[12 < k ≤ 13, ((k - 12) * (θ - ε)), θ - ε], C) → g_2
  101. Segment(A, B) → h_2