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正方形、切线与两相切圆

正方形、切线与两相切圆
几何画板及绘图内容加载中，请稍等……
2016-12-09 10:33
圆多边形向量解析几何
①本范例演示正方形、两相切圆及它们公共外切线的几何关系；②涉及核心数学概念包括圆、多边形、角、向量、函数、解析几何、几何作图等；③构造过程中运用了Circle（圆）、Segment（线段）、Intersect（交点）、Angle（角度）、Dilate（缩放）、Rotate（旋转）、Line（直线）、Midpoint（中点）、Polygon（多边形）、Vector（向量）、Translate（平移）等GeoGebra命令与对象；④适合高中阶段学生学习和理解几何构造与性质，可用于解析几何和几何作图的教学。

GeoGebra 命令及参数

  1. Circle(A, D) → c
  2. Circle(B, F_2) → d
  3. Segment(H_1, G_1) → f
  4. Circle(G_1, f) → e
  5. Circle(H_1, f) → g
  6. Intersect(c, g, 1) → M
  7. Intersect(d, e, 2) → H
  8. Angle(M, H, G_1) → α
  9. Angle(H, G_1, E_2) → β
  10. Angle(G_1, H_1, M) → γ
  11. Angle(H_1, M, H) → δ
  12. Segment(M, H) → h
  13. Segment(H, G_1) → i
  14. Segment(M, H_1) → j
  15. Dilate(A, If[0 < b ≤ 1, 1 - b, 0], D) → A'
  16. Segment(A, A') → l
  17. Rotate(A, If[1 < b ≤ 2, ((b - 1) * π / 2), π / 2], A) → A'_1
  18. Rotate(A', If[1 < b ≤ 2, ((b - 1) * π / 2), π / 2], A) → A''
  19. Segment(A'_1, A'') → l'
  20. Rotate(A'_1, If[2 < b ≤ 3, ((b - 2) * π), π], C) → K
  21. Rotate(A'', If[2 < b ≤ 3, ((b - 2) * π), π], C) → A'''
  22. Segment(K, A''') → l''
  23. Line(B, F_2) → m
  24. Circle(F_2, l / 5) → p
  25. Intersect(p, m) → L, M_1
  26. Circle(L, l / 5) → q
  27. Intersect(q, m) → N, O
  28. Circle(N, l / 5) → r
  29. Intersect(r, m) → P, Q
  30. Midpoint(P, B) → R
  31. Midpoint(M, H) → S
  32. Dilate(M, If[4 < b ≤ 5, 5 - b, 0], S) → I'
  33. Segment(M, I') → n
  34. Dilate(S, If[5 < b ≤ 6, 6 - b, 0], H) → S'
  35. Segment(S, S') → s
  36. Rotate(M, If[7 < b ≤ 8, ((b - 7) * π), π], S) → I'_1
  37. Rotate(I', If[7 < b ≤ 8, ((b - 7) * π), π], S) → I''
  38. Rotate(M, If[6 < b ≤ 7, ((b - 6) * π), π], S) → I'_2
  39. Rotate(I', If[6 < b ≤ 7, ((b - 6) * π), π], S) → I''_1
  40. Segment(I'_2, I''_1) → n'
  41. Rotate(M, If[7 < b ≤ 8, ((b - 7) * π / 2), π / 2], H) → G
  42. Rotate(I', If[7 < b ≤ 8, ((b - 7) * π / 2), π / 2], H) → I''_2
  43. Segment(G, I''_2) → n'_1
  44. Rotate(I'_2, If[7 < b ≤ 8, ((b - 7) * π / 2), π / 2], H) → T
  45. Rotate(I''_1, If[7 < b ≤ 8, ((b - 7) * π / 2), π / 2], H) → U
  46. Segment(T, U) → n''
  47. Rotate(S, If[7 < b ≤ 8, ((b - 7) * π / 2), π / 2], H) → S'_1
  48. Angle(M, H, G_1) → ε
  49. Rotate(T, If[8 < b ≤ 9, ((b - 8) * π / 2), π / 2], G_1) → F
  50. Rotate(U, If[8 < b ≤ 9, ((b - 8) * π / 2), π / 2], G_1) → U'
  51. Segment(F, U') → n'''
  52. Rotate(G, If[8 < b ≤ 9, ((b - 8) * π / 2), π / 2], G_1) → V
  53. Rotate(I''_2, If[8 < b ≤ 9, ((b - 8) * π / 2), π / 2], G_1) → W
  54. Segment(V, W) → t
  55. Rotate(S'_1, If[8 < b ≤ 9, ((b - 8) * π / 2), π / 2], G_1) → Z
  56. Angle(H, G_1, H_1) → ζ
  57. Rotate(F, If[9 < b ≤ 10, ((b - 9) * π / 2), π / 2], H_1) → T''
  58. Rotate(U', If[9 < b ≤ 10, ((b - 9) * π / 2), π / 2], H_1) → U''
  59. Segment(T'', U'') → f_1
  60. Rotate(Z, If[9 < b ≤ 10, ((b - 9) * π / 2), π / 2], H_1) → Z'
  61. Rotate(V, If[9 < b ≤ 10, ((b - 9) * π / 2), π / 2], H_1) → V'
  62. Rotate(W, If[9 < b ≤ 10, ((b - 9) * π / 2), π / 2], H_1) → W'
  63. Segment(V', W') → t'
  64. Polygon(H_1, G_1, H, M) → poly1, h_1, g_1, j_1, i_1
  65. Angle(G_1, H_1, M) → η
  66. Dilate(H_1, If[10 < b ≤ 11, 11 - b, 0], D) → H'
  67. Dilate(G_1, If[10 < b ≤ 11, 11 - b, 0], F_2) → E
  68. Segment(G_1, E) → k_1
  69. Segment(H_1, H') → l_1
  70. Angle(H_1, D, A) → θ
  71. Dilate(F_2, If[11 < b ≤ 12, 12 - b, 0], N) → F'
  72. Segment(F_2, F') → m_1
  73. Dilate(N, If[12 < b ≤ 13, 13 - b, 0], R) → N'
  74. Segment(N, N') → n_1
  75. Dilate(R, If[13 < b ≤ 14, 14 - b, 0], B) → R'
  76. Segment(R, R') → p_1
  77. Dilate(H, If[14 < b ≤ 15, 15 - b, 0], N) → J'
  78. Vector(H, J') → u
  79. Translate(T, u) → T'_1
  80. Translate(U, u) → U'_1
  81. Segment(T'_1, U'_1) → n'''_1
  82. Translate(S'_1, u) → A_1
  83. Translate(G, u) → B_1
  84. Translate(I''_2, u) → C_1
  85. Segment(B_1, C_1) → q_1
  86. Dilate(H, If[15 < b ≤ 16, 16 - b, 0], R) → J'_1
  87. Vector(H, J'_1) → v
  88. Translate(S'_1, v) → D_1
  89. Translate(T, v) → T'_2
  90. Translate(U, v) → U'_2
  91. Segment(T'_2, U'_2) → n'''_2
  92. Translate(G, v) → E_1
  93. Translate(I''_2, v) → F_1
  94. Segment(E_1, F_1) → r_1
  95. Dilate(H, If[16 < b ≤ 17, 17 - b, 0], B) → J'_2
  96. Vector(H, J'_2) → w
  97. Translate(T, w) → T'_3
  98. Translate(U, w) → U'_3
  99. Segment(T'_3, U'_3) → n'''_3
  100. Angle(B, F_2, G) → ι