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垂心与三角形外接圆动态构造

垂心与三角形外接圆动态构造
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2016-10-21 06:40
测试改名0圆几何作图轨迹
该范例演示了三角形垂心的几何构造及其与外接圆的关系。核心数学概念包括三角形的垂心（三条高线的交点）、外接圆（过三角形三个顶点的圆）以及高线的垂直性质。作图要点：使用GeoGebra的Segment、Point、Circle、Intersect、OrthogonalLine、Line等命令，依次构造三角形ABC，通过作各边的垂线（OrthogonalLine）并求交点得到垂心D，再构造外接圆c，并利用垂心与外接圆上的点生成更多相关对象（如垂足、延长线等），形成动态可交互的图形。适合中学几何教学，帮助学生直观理解垂心与外接圆的性质，也可用于探究垂心在不同三角形类型（锐角、直角、钝角）中的位置变化。

GeoGebra 命令及参数

  1. Segment(A, B) → f
  2. Segment(B, C) → g
  3. Segment(A, C) → h
  4. Point(h) → D
  5. Circle(B, D) → c
  6. Intersect(c, h, 2) → E
  7. Point(h) → F
  8. Circle(E, F) → d
  9. Segment(A_1, B_1) → f_1
  10. Segment(B_1, C_1) → g_1
  11. Segment(A_1, C_1) → h_1
  12. Point(h_1) → D_1
  13. Circle(B_1, D_1) → c_1
  14. Intersect(c_1, h_1, 2) → E_1
  15. Segment(A_2, B_2) → f_2
  16. Segment(B_2, C_2) → g_2
  17. Segment(A_2, C_2) → h_2
  18. Point(h_2) → D_2
  19. Circle(B_2, D_2) → c_2
  20. Intersect(c_2, h_2, 2) → E_2
  21. Point(h_2) → F_1
  22. Circle(E_2, F_1) → d_1
  23. Intersect(d, d_1, 1) → G
  24. Intersect(d, d_1, 2) → H
  25. Segment(B, H) → i
  26. Segment(H, B) → j
  27. OrthogonalLine(F, i) → k
  28. Intersect(k, h) → I
  29. Intersect(k, h) → J
  30. Point(i) → K
  31. OrthogonalLine(K, h) → l
  32. Point(f) → L
  33. Line(B, L) → m
  34. Circle(F_1, M) → e
  35. Intersect(e, m, 2) → N
  36. Intersect(e, m, 1) → O
  37. Segment(A_3, B_3) → f_3
  38. Segment(B_3, C_3) → g_3
  39. Segment(A_3, C_3) → h_3
  40. Point(h_3) → D_3
  41. Circle(B_3, D_3) → c_3
  42. Point(h_3) → F_2
  43. Segment(A_4, B_4) → f_4
  44. Segment(B_4, C_4) → g_4
  45. Segment(A_4, C_4) → h_4
  46. Point(h_4) → D_4
  47. Circle(B_4, D_4) → c_4
  48. Point(h_4) → F_3
  49. Point(c_4) → P
  50. Point(c_4) → Q
  51. Segment(F_1, Q) → n
  52. Segment(Q, P) → p
  53. Segment(F_1, P) → q
  54. Intersect(d_1, q, 2) → R
  55. Line(R, P) → r
  56. Point(l) → S
  57. Point(l) → T
  58. Line(S, T) → s
  59. Point(g) → U
  60. Point(g) → V
  61. Line(U, V) → t
  62. Segment(A_5, C_5) → h_5
  63. Point(h_5) → D_5
  64. Point(h_5) → F_4
  65. Circle(F_4, M_1) → e_1
  66. Segment(A_6, B_5) → f_5
  67. Segment(B_5, C_6) → g_5
  68. Segment(A_6, C_6) → h_6
  69. Point(h_6) → D_6
  70. Circle(B_5, D_6) → c_5
  71. Point(h_6) → F_5
  72. Segment(A_7, B_7) → f_6
  73. Segment(B_7, C_7) → g_6
  74. Segment(A_7, C_7) → h_7
  75. Point(h_7) → D_7
  76. Circle(B_7, D_7) → c_6
  77. Point(h_7) → F_6
  78. Point(c_5) → W
  79. Point(c_5) → Z
  80. Point(c_5) → G_1
  81. Line(W, G_1) → a
  82. Intersect(l, r) → H_1
  83. Point(a) → I_1
  84. Segment(I_1, H_1) → b
  85. Segment(F_1, I_1) → i_1
  86. Segment(H_1, F_1) → j_1
  87. Segment(I_1, H_1) → k_1
  88. Segment(K, H_1) → l_1
  89. Segment(B, F_4) → m_1