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内切圆与外接圆关系动态演示

内切圆与外接圆关系动态演示
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2017-04-22 23:33
测试改名0圆三角函数几何作图
①本范例动态演示三角形内切圆与其中一个外接圆的关系；②涉及核心数学概念包括圆、三角形、角度、几何变换等；③构造中运用了旋转、射线、点、线段、最小值、角度、圆、交点、圆扇形、垂线、中点、多边形、缩放、圆弧、向量、平移、直线等GeoGebra命令与对象；④适合高中及以上数学学习者，用于理解三角形与圆的几何性质及动态关系。

GeoGebra 命令及参数

  1. Rotate(B, (a * π / 180), A) → B'
  2. Ray(A, B') → g
  3. Point(g) → C
  4. Segment(A, C) → j
  5. Ray(A, B) → f
  6. Point(f) → D
  7. Segment(C, D) → h
  8. Segment(A, D) → i
  9. Min({i, j, h}) → b
  10. Angle(B, A, B') → α
  11. Circle(D, (0.2 * b)) → d
  12. Circle(C, 4) → c
  13. Intersect(c, g, 2) → E
  14. Intersect(d, f, 2) → F
  15. Ray(D, F) → k
  16. Ray(C, E) → l
  17. Circle(C, (0.2 * b)) → e
  18. Circle(D, (0.15 * b)) → p
  19. Angle(F, D, C) → β
  20. Angle(D, C, E) → γ
  21. Rotate(F, β / 2, D) → F'
  22. Angle(F, D, F') → δ
  23. Rotate(D, γ / 2, C) → D'
  24. Angle(D, C, D') → ε
  25. Ray(C, D') → m
  26. Ray(D, F') → n
  27. Intersect(e, m, 1) → G
  28. Intersect(e, h, 1) → H
  29. Intersect(p, h, 1) → I
  30. CircleSector(C, H, G) → q
  31. CircleSector(D, F, F') → r
  32. Intersect(m, n) → J
  33. OrthogonalLine(J, k) → s
  34. OrthogonalLine(J, l) → t
  35. Intersect(s, k) → K
  36. Intersect(t, l) → L
  37. Circle(J, K) → c_1
  38. Segment(L, K) → f_1
  39. Midpoint(f_1) → M
  40. OrthogonalLine(M, f_1) → g_1
  41. Intersect(c_1, g_1, 2) → N
  42. Segment(N, C) → h_1
  43. Segment(N, D) → i_1
  44. Polygon(A, D, C) → poly1, c_2, a_1, d_2
  45. Angle(J, K, D) → ζ
  46. Angle(C, L, J) → η
  47. Dilate(A, If[0 < o ≤ 1, 1 - o, 0], D) → A'
  48. Dilate(A, If[0 < o ≤ 1, 1 - o, 0], C) → A'_1
  49. Segment(A, A') → k_1
  50. Segment(A, A'_1) → l_1
  51. Dilate(C, If[1 < o ≤ 2, 2 - o, 0], J) → C'
  52. Dilate(D, If[1 < o ≤ 2, 2 - o, 0], J) → D'_1
  53. Segment(C, C') → m_1
  54. Segment(D, D'_1) → n_1
  55. Rotate(r, If[2.5 < o ≤ 3.5, ((o - 2.5) * δ), δ], D) → r'
  56. Rotate(q, If[3.5 < o ≤ 4.5, ((o - 3.5) * ε), ε], C) → q'
  57. Dilate(J, If[5 < o ≤ 6, 6 - o, 0], L) → J'
  58. Dilate(J, If[5 < o ≤ 6, 6 - o, 0], K) → J'_1
  59. Segment(J, J'_1) → p_1
  60. Segment(J, J') → q_1
  61. Angle(L, J, K) → θ
  62. Rotate(L, If[6 < o ≤ 7, ((o - 6) * θ), θ], J) → L'
  63. CircleArc(J, L, L') → e_1
  64. Rotate(K, If[7 < o ≤ 8, ((o - 7) * ((2 * π) - θ) / 2), ((2 * π) - θ) / 2], J) → K'
  65. CircleArc(J, K, K') → r_1
  66. Rotate(N, If[8 < o ≤ 9, ((o - 8) * ((2 * π) - θ) / 2), ((2 * π) - θ) / 2], J) → N'
  67. CircleArc(J, N, N') → s_1
  68. Dilate(N, If[11 < o ≤ 12, 12 - o, 0], D) → N'_1
  69. Dilate(N, If[11 < o ≤ 12, 12 - o, 0], C) → N'_2
  70. Min(c_2, d_2) → w
  71. Circle(A, (0.15 * w)) → e_2
  72. Segment(A, C) → m_2
  73. Intersect(e_2, m_2) → S,
  74. Segment(A, D) → n_2
  75. Intersect(e_2, n_2) → T,
  76. Min({h, k_1, l_1}) → b_1
  77. Intersect(d, h, 1) → U
  78. Intersect(e, h) → W,
  79. Angle(A, C, D) → λ
  80. Angle(C, D, A) → κ
  81. Rotate(A, λ / 2, C) → A'_2
  82. Angle(A, C, A'_2) → ι
  83. Ray(C, A'_2) → t_1
  84. Rotate(C, κ / 2, D) → C'_1
  85. Angle(C, D, C'_1) → μ_1
  86. Ray(D, C'_1) → f_2
  87. Intersect(d, f_2) → A_1,
  88. CircleSector(D, U, A_1) → r_2
  89. Intersect(t_1, f_2) → Z
  90. OrthogonalLine(Z, k_1) → s_2
  91. OrthogonalLine(Z, l_1) → t_2
  92. Intersect(s_2, k_1) → B_1
  93. Intersect(t_2, l_1) → C_1
  94. Intersect(k_1, t_1) → D_1
  95. Circle(A, (0.2 * b)) → q_2
  96. Angle(D, A, C) → μ
  97. Rotate(D, μ / 2, A) → D'_2
  98. Angle(D, A, D'_2) → ν
  99. Ray(A, D'_2) → a_2
  100. Intersect(q_2, a_2) → V,
  101. Intersect(q_2, k_1) → E_1,
  102. CircleSector(A, E_1, V) → p_2
  103. Dilate(D, If[1 < o ≤ 2, 2 - o, 0], Z) → D'_3
  104. Dilate(A, If[1 < o ≤ 2, 2 - o, 0], Z) → A'_3
  105. Segment(A, A'_3) → b_2
  106. Segment(D, D'_3) → f_3
  107. Rotate(r_2, If[2.5 < o ≤ 3.5, ((o - 2.5) * κ / 2), κ / 2], D) → c_3
  108. Rotate(p_2, If[3.5 < o ≤ 4.5, ((o - 3.5) * μ / 2), μ / 2], A) → d_3
  109. Dilate(Z, If[5 < o ≤ 6, 6 - o, 0], C_1) → Z'
  110. Dilate(Z, If[5 < o ≤ 6, 6 - o, 0], B_1) → Z'_1
  111. Segment(Z, Z') → g_3
  112. Segment(Z, Z'_1) → h_3
  113. Angle(Z, B_1, A) → ξ
  114. Angle(A, C_1, Z) → ο
  115. Circle(Z, Z') → e_3
  116. Rotate(C_1, If[6 < o ≤ 9, (((o - 6) * 2) * π / 3), (2 * π)], Z) → F_1
  117. Segment(C, D_1) → i_3
  118. Intersect(e, i_3) → G_1,
  119. Intersect(e, l_1) → H_1,
  120. Dilate(C, If[9.5 < o ≤ 10.5, 10.5 - o, 0], Z) → C'_2
  121. Dilate(J, If[9.5 < o ≤ 10.5, 10.5 - o, 0], Z) → J'_2
  122. Segment(J, J'_2) → j_1
  123. Segment(C, C'_2) → d_1
  124. CircleSector(C, H_1, G_1) → g_2
  125. Polygon(A, Z, C) → poly2, c_4, a_3, z_1
  126. Polygon(A, D, J) → poly3, j_2, a_4, d_4
  127. CircleArc(A, E_1, V) → h_2
  128. CircleArc(Z, Z', F_1) → k_3
  129. Dilate(C, If[11 < o ≤ 12, 12 - o, 0], J) → C'_3
  130. Vector(C, C'_3) → u_1
  131. Translate(g_2, u_1) → k_2
  132. Line(C, j_1) → i_2
  133. Point(i_2) → O
  134. Angle(O, C, H_1) → ρ
  135. Rotate(k_2, If[12 < o ≤ 13, ((-(o - 12)) * ρ), (-ρ)], J) → p_3

AI 绘图提示语参考

画一个三角形，动态展示其内切圆与一个外切圆的关系。首先画出三角形的一边，然后在该边上找到中点，以此中点为圆心画一个内切圆。接着，从内切圆的圆心向三角形的一角画一条线段，以此线段为半径画一个外切圆。再从内切圆的圆心向三角形的另一角画一条线段，并以此线段为半径画另一个外切圆。最后，将内切圆和外切圆的交点连接起来，形成动态的几何关系。
以上提示语为根据原始绘图命令整理而来，仅供参考。