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对称轴探究：多边形与圆的镜像对称

对称轴探究：多边形与圆的镜像对称
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2017-11-29 14:30
圆多边形角变换几何作图
本范例通过动态几何演示多边形及其内切圆、外接圆等图形的对称轴性质。核心数学概念包括轴对称、镜像变换、角平分线、垂直平分线以及对称轴与图形元素（如顶点、边、圆心）的关系。作图要点：使用GeoGebra命令Polygon创建多边形，OrthogonalLine和LineBisector构造垂线与中垂线，AngularBisector作角平分线，Mirror实现镜像对称，Intersect求交点，Circle和Incircle构造圆及内切圆，Segment连接线段，Length测量长度，Midpoint取中点，Point定义点。通过拖动顶点或调整参数，可动态观察对称轴数量与位置的变化。适合初中至高中阶段学习几何对称、变换及图形性质，也可用于教师课堂演示或学生自主探究。

GeoGebra 命令及参数

  1. Polygon(A, B, 3) → poly1, f, g, h, C
  2. OrthogonalLine(C, f) → i
  3. OrthogonalLine(A, g) → j
  4. Intersect(i, j) → D
  5. Line(E, F) → k
  6. Circle(D, Segment[A, B]) → c
  7. Intersect(c, k, 1) → G
  8. Intersect(c, k, 2) → H
  9. Segment(H, G) → l
  10. Mirror(poly1, l) → poly1'
  11. Polygon(I, J, K) → t1, k_1, i_1, j_1
  12. AngularBisector(J, I, K) → m
  13. AngularBisector(I, K, J) → n
  14. Intersect(m, n) → L
  15. Length(i_1) → a
  16. Length(j_1) → b
  17. Length(k_1) → d
  18. Circle(L, Max[a, Max[b, d]]) → e
  19. Line(M, N) → p
  20. Intersect(e, p, 1) → O
  21. Intersect(e, p, 2) → P
  22. Segment(O, P) → q
  23. Mirror(I, q) → I'
  24. Mirror(J, q) → J'
  25. Mirror(K, q) → K'
  26. Polygon(I', J', K') → t1', k', i', j'
  27. Polygon(Q, R, 4) → poly2, s, t, f_1, g_1, S, T
  28. Midpoint(Q, T) → U
  29. Midpoint(T, S) → V
  30. OrthogonalLine(U, g_1) → h_1
  31. OrthogonalLine(V, f_1) → l_1
  32. Point(l_1) → W
  33. Circle(W, Segment[R, Q]) → c_1
  34. Line(Z, A_1) → m_1
  35. Intersect(c_1, m_1, 1) → B_1
  36. Intersect(c_1, m_1, 2) → C_1
  37. Segment(B_1, C_1) → n_1
  38. Mirror(poly2, n_1) → poly2'
  39. Segment(D_1, E_1) → p_1
  40. OrthogonalLine(E_1, p_1) → q_1
  41. Circle(E_1, Segment[E_1, D_1]) → d_1
  42. Intersect(d_1, q_1, 2) → F_1
  43. Circle(F_1, Segment[E_1, D_1]) → e_1
  44. Intersect(e_1, q_1, 2) → G_1
  45. OrthogonalLine(G_1, q_1) → r_1
  46. OrthogonalLine(D_1, p_1) → s_1
  47. Intersect(r_1, s_1) → H_1
  48. Polygon(E_1, G_1, H_1, D_1) → q1, e_2, g_2, h_2, d_2
  49. LineBisector(D_1, E_1) → t_1
  50. LineBisector(e_2) → a_1
  51. Point(t_1) → I_1
  52. Circle(I_1, e_2) → c_2
  53. Line(J_1, K_1) → b_1
  54. Intersect(c_2, b_1, 1) → L_1
  55. Intersect(c_2, b_1, 2) → M_1
  56. Segment(L_1, M_1) → f_2
  57. Segment(L_1, M_1) → i_2
  58. Mirror(E_1, i_2) → N_1
  59. Mirror(G_1, i_2) → O_1
  60. Mirror(H_1, i_2) → P_1
  61. Mirror(D_1, i_2) → Q_1
  62. Polygon(N_1, O_1, P_1, Q_1) → q1', n_2, o_1, p_2, q_2
  63. Line(R_1, e_2) → j_2
  64. OrthogonalLine(R_1, j_2) → k_2
  65. Circle(R_1, Segment[E_1, D_1]) → r_2
  66. Intersect(r_2, k_2, 2) → T_1
  67. Line(T_1, j_2) → l_2
  68. Line(R_1, S_1) → m_2
  69. Point(j_2) → U_1
  70. Line(U_1, m_2) → s_2
  71. Intersect(l_2, s_2) → V_1
  72. Intersect(l_2, m_2) → W_1
  73. Polygon(V_1, W_1, R_1, U_1) → q2, v_1, w_1, r_3, u_1
  74. Midpoint(W_1, U_1) → Z_1
  75. Circle(Z_1, Segment[R_1, V_1]) → t_2
  76. Line(A_2, B_2) → a_2
  77. Intersect(t_2, a_2, 2) → C_2
  78. Intersect(t_2, a_2, 1) → D_2
  79. Segment(C_2, D_2) → b_2
  80. Mirror(V_1, b_2) → E_2
  81. Mirror(W_1, b_2) → F_2
  82. Mirror(R_1, b_2) → G_2
  83. Mirror(U_1, b_2) → H_2
  84. Polygon(E_2, F_2, G_2, H_2) → q2', e_3, f_3, g_3, h_3