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三角形内心与内切圆的动态演示

三角形内心与内切圆的动态演示
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2016-05-12 03:17
测试改名0圆角几何作图滑块动画
该范例通过GeoGebra动态演示三角形内心（内切圆圆心）的性质，直观展示内心到三边距离相等、内切圆与三边相切等核心几何关系。涉及的核心数学概念包括三角形内心、内切圆、角平分线、距离与比例、正弦定理等。作图要点：利用GeoGebra的Polygon命令构造三角形，通过Angle、Line、Intersect等命令确定角平分线交点（内心），再使用Incircle命令生成内切圆，并借助Circle、Segment、Rotate、Dilate等命令构造辅助线、半径和动态角度，通过滑块或拖动顶点实时观察图形变化。适合中学几何教学、数学实验或自学，帮助学生理解内心与内切圆的定义及性质，并培养动态几何探究能力。

GeoGebra 命令及参数

  1. Line(A, B) → d
  2. Slope(d) → a
  3. PointIn(c) → C
  4. Line(A, C) → e
  5. Line(B, C) → f
  6. Angle(B, A, C) → α
  7. Angle(A, C, B) → β
  8. Angle(C, B, A) → γ
  9. Segment(A, C) → g
  10. Segment(C, B) → h
  11. Segment(B, A) → i
  12. Min(list1) → j
  13. Circle(C, (0.2 * j)) → q
  14. Circle(A, (0.2 * j)) → p
  15. Circle(B, (0.2 * j)) → k
  16. Intersect(q, f, 2) → D
  17. Intersect(q, e, 1) → E
  18. Intersect(q, e, 2) → F
  19. Intersect(p, e, 2) → G
  20. Intersect(p, d, 1) → H
  21. Intersect(k, f, 2) → I_2
  22. Intersect(k, d, 2) → J
  23. Intersect(d, i) → K
  24. Intersect(d, i) → L
  25. Intersect(k, f, 1) → N
  26. Intersect(p, e, 1) → O
  27. Intersect(p, d, 2) → P
  28. Angle(D, C, E) → δ
  29. Angle(O, A, P) → ε
  30. Angle(J, B, I_2) → η
  31. Angle(G, A, H) → θ
  32. Incircle(A, B, C) → r
  33. Center(r) → I
  34. Line(C, I) → l
  35. Line(A, I) → n
  36. Line(B, I) → s
  37. Intersect(e, s) → R
  38. Intersect(d, l) → S
  39. Intersect(f, n) → T
  40. Polygon(A, B, C) → poly1, c_1, a_1, b_1
  41. Rotate(D, δ / 2, C) → D'
  42. Angle(D, C, D') → κ
  43. Rotate(J, η / 2, B) → J'
  44. Angle(J, B, J') → λ
  45. Ray(C, D') → t
  46. Ray(B, J') → e_1
  47. Rotate(O, ε / 2, A) → O'
  48. Angle(O, A, O') → ι
  49. Ray(A, O') → d_1
  50. Segment(C, S) → f_1
  51. Segment(A, T) → g_1
  52. Segment(B, R) → h_1
  53. Intersect(q, f_1, 1) → U
  54. Intersect(q, f, 1) → V
  55. Intersect(p, g_1, 1) → W
  56. Intersect(k, h_1, 1) → Z
  57. CircleSector(C, E, U) → k_1
  58. CircleSector(A, P, W) → q_1
  59. CircleSector(B, I_2, Z) → p_1
  60. Dilate(C, If[0 < m ≤ 1, 1 - m, 0], S) → C'
  61. Segment(C, C') → i_1
  62. Rotate(k_1, If[1 < m ≤ 2, ((m - 1) * β / 2), β / 2], C) → r_1
  63. Dilate(B, If[2 < m ≤ 3, 3 - m, 0], R) → B'
  64. Segment(B, B') → j_1
  65. Rotate(p_1, If[3 < m ≤ 4, ((m - 3) * γ / 2), γ / 2], B) → s_1
  66. Dilate(A, If[4 < m ≤ 5, 5 - m, 0], T) → A'
  67. Segment(A, A') → l_1
  68. Rotate(q_1, If[5 < m ≤ 6, ((m - 5) * α / 2), α / 2], A) → t_1
  69. OrthogonalLine(I, f) → m_1
  70. OrthogonalLine(I, d) → n_1
  71. OrthogonalLine(I, e) → a_2
  72. Intersect(f, m_1) → M
  73. Intersect(d, n_1) → A_1
  74. Intersect(e, a_2) → B_1
  75. Segment(I, M) → b_2
  76. Circle(M, (0.2 * b_2)) → c_2
  77. Intersect(c_2, f, 1) → C_1
  78. Intersect(c_2, b_2, 1) → D_1
  79. Intersect(c_2, f, 2) → E_1
  80. CircleSector(M, E_1, D_1) → d_2
  81. Dilate(I, If[6 < m ≤ 7, 7 - m, 0], M) → Q'
  82. Segment(I, Q') → e_2
  83. Rotate(d_2, If[7.5 < m ≤ 8.5, ((m - 7.5) * π / 2), 1.5707963267948966], M) → f_2
  84. Segment(I, B_1) → g_2
  85. Segment(I, A_1) → h_2
  86. Circle(B_1, (0.2 * g_2)) → k_2
  87. Circle(A_1, (0.2 * h_2)) → p_2
  88. Intersect(k_2, e, 2) → F_1
  89. Intersect(k_2, g_2, 1) → G_1
  90. Intersect(k_2, e, 1) → H_1
  91. Intersect(p_2, d, 2) → I_1
  92. Intersect(p_2, h_2, 1) → J_1
  93. Intersect(p_2, d, 1) → K_1
  94. Dilate(I, If[8.5 < m ≤ 9.5, 9.5 - m, 0], B_1) → Q'_1
  95. Segment(I, Q'_1) → i_2
  96. CircleSector(B_1, H_1, G_1) → q_2
  97. CircleSector(A_1, I_1, J_1) → r_2
  98. Rotate(q_2, If[10 < m ≤ 11, ((m - 10) * π / 2), 1.5707963267948966], B_1) → s_2
  99. Dilate(I, If[11 < m ≤ 12, 12 - m, 0], A_1) → Q'_2
  100. Segment(I, Q'_2) → j_2
  101. Rotate(r_2, If[12.5 < m ≤ 13.5, ((m - 12.5) * π / 2), 1.5708], A_1) → t_2
  102. Angle(M, I, B_1) → ζ
  103. Angle(B_1, I, A_1) → μ
  104. Angle(A_1, I, M) → ν
  105. Rotate(I, If[14 < m ≤ 15, ((m - 14) * ζ), ζ], I) → Q'_3
  106. Rotate(Q', If[14 < m ≤ 15, ((m - 14) * ζ), ζ], I) → Q''
  107. Segment(Q'_3, Q'') → l_2
  108. Rotate(Q'_3, If[15 < m ≤ 16, ((m - 15) * μ), μ], I) → L_1
  109. Rotate(Q'', If[15 < m ≤ 16, ((m - 15) * μ), μ], I) → Q'''
  110. Segment(L_1, Q''') → m_2
  111. Rotate(A_1, If[16 < m ≤ 18, ((m - 16) * π), 6.28319], I) → M'
  112. Intersect(m_1, f) → Q
  113. Circle(I, Q) → d_3
  114. Line(A, B) → n_2
  115. CircleArc(I, A_1, M') → c_3