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Vivani定理演示

Vivani定理演示
几何画板及绘图内容加载中，请稍等……
2016-05-21 10:15
圆立体几何几何作图
本范例通过无文字、无线段长度、无角度度量的方式，生动展示了Vivani定理。它涉及圆、球体和特殊点等核心数学概念。通过使用Polygon、Dilate、Circle、Intersect等GeoGebra命令，构建了一个几何图形。适合高中数学教学和学习。

GeoGebra 命令及参数

  1. Polygon(A, B, 3) → poly1, a, b, c, C
  2. Dilate(A, If[0 < d ≤ 1, 1 - d, 0], B) → A'
  3. Segment(A, A') → e
  4. Dilate(B, If[1 < d ≤ 2, 2 - d, 0], C) → B'
  5. Segment(B, B') → f
  6. Dilate(C, If[2 < d ≤ 3, 3 - d, 0], A) → C'
  7. Segment(C, C') → g
  8. Segment(A, B) → h
  9. Segment(A, C) → i
  10. Segment(B, C) → j
  11. Circle(A, (0.15 * h)) → k
  12. Intersect(k, i, 1) → D
  13. Intersect(k, h, 1) → E
  14. Rotate(A, If[3.5 < d ≤ 4.5, ((d - 3.5) * π / 3), 1.0471975511965976], A) → A'_1
  15. Rotate(A', If[3.5 < d ≤ 4.5, ((d - 3.5) * π / 3), 1.0471975511965976], A) → A''
  16. Segment(A'_1, A'') → e'
  17. Rotate(A, If[4.5 < d ≤ 5.5, ((-(d - 4.5)) * π / 3), -1.0471975511965976], B) → A'_2
  18. Rotate(A', If[4.5 < d ≤ 5.5, ((-(d - 4.5)) * π / 3), -1.0471975511965976], B) → A''_1
  19. Segment(A'_2, A''_1) → e'_1
  20. Dilate(A, If[6 < d ≤ 7, 7 - d, 0], B) → A'_3
  21. Vector(A, A'_3) → u
  22. Circle(B, (0.15 * h)) → r
  23. Intersect(f, e'_1) → G
  24. Intersect(f, e'_1) → H
  25. Intersect(f, e'_1) → I
  26. Intersect(f, e'_1) → J
  27. Intersect(r, j, 1) → F
  28. Intersect(r, h, 1) → K
  29. Dilate(B, If[8 < d ≤ 9, 9 - d, 0], C) → B'_1
  30. Vector(B, B'_1) → v
  31. Intersect(g, e') → L
  32. Midpoint(h) → O
  33. Circle(O, (0.05 * h)) → d_1
  34. Segment(C, O) → m
  35. Intersect(d_1, m, 1) → P
  36. Intersect(d_1, h) → Q, R
  37. CircleSector(O, R, P) → f_1
  38. Polygon(A, B, C) → poly2, c_2, a_1, b_1
  39. PointIn(poly2) → T
  40. OrthogonalLine(T, h) → n
  41. OrthogonalLine(T, i) → g_1
  42. OrthogonalLine(T, j) → h_1
  43. Intersect(g_1, i) → U
  44. Intersect(n, h) → V
  45. Intersect(h_1, j) → W
  46. Angle(T, U, C) → α
  47. Angle(T, V, A) → β
  48. Angle(T, W, B) → γ
  49. Circle(W, (0.05 * h)) → q
  50. Circle(V, (0.05 * h)) → s
  51. Circle(U, (0.05 * h)) → t
  52. Intersect(q, j, 2) → Z
  53. Intersect(q, h_1, 2) → A_1
  54. Intersect(s, h, 2) → B_1
  55. Intersect(s, n, 2) → C_1
  56. Intersect(t, i, 1) → D_1
  57. Intersect(t, g_1, 1) → E_1
  58. CircleSector(W, Z, A_1) → k_1
  59. CircleSector(V, B_1, C_1) → p_1
  60. CircleSector(U, D_1, E_1) → q_1
  61. Dilate(T, If[10 < d ≤ 11, 11 - d, 0], U) → T'
  62. Segment(T, T') → i_1
  63. Segment(T, U) → j_1
  64. Segment(T, W) → l_1
  65. Segment(T, V) → m_1
  66. Circle(C, j_1) → c_1
  67. Segment(C, O) → e_1
  68. Intersect(c_1, e_1, 1) → M
  69. Circle(M, m_1) → s_1
  70. Intersect(s_1, e_1, 1) → N
  71. Dilate(C, If[5.5 < d ≤ 6.5, 6.5 - d, 0], M) → C'_1
  72. Segment(C, C'_1) → l
  73. Dilate(M, If[6.5 < d ≤ 7.5, 7.5 - d, 0], N) → M'
  74. Segment(M, M') → n_1
  75. Dilate(N, If[7.5 < d ≤ 8.5, 8.5 - d, 0], O) → N'
  76. Segment(N, N') → t_1
  77. Rotate(f_1, If[9 < d ≤ 10, ((d - 9) * π / 2), 1.5708], O) → p
  78. Dilate(T, If[10 < d ≤ 11, 11 - d, 0], V) → T'_1
  79. Dilate(T, If[10 < d ≤ 11, 11 - d, 0], W) → T'_2
  80. Segment(T, T'_1) → a_2
  81. Segment(T, T'_2) → b_2
  82. Rotate(q_1, If[11.5 < d ≤ 12.5, ((d - 11.5) * π / 2), 1.5708], U) → r_1
  83. Rotate(p_1, If[11.5 < d ≤ 12.5, ((d - 11.5) * π / 2), 1.5708], V) → d_2
  84. Rotate(k_1, If[11.5 < d ≤ 12.5, ((d - 11.5) * π / 2), 1.5708], W) → e_2
  85. Dilate(U, If[13 < d ≤ 14, 14 - d, 0], C) → U'
  86. Vector(U, U') → w
  87. Translate(T, w) → T'_3
  88. Translate(T', w) → T''
  89. Segment(T'_3, T'') → f_2
  90. Rotate(T'_3, If[14 < d ≤ 15, (((d - 14) * 2) * π / 3), 2.0943951023931953], M) → S
  91. Rotate(T'', If[14 < d ≤ 15, (((d - 14) * 2) * π / 3), 2.0943951023931953], M) → T'''
  92. Rotate(T'_3, If[14 < d ≤ 15, d - 14 - π / 3, -1.0471975511965976], C) → F_1
  93. Rotate(T'', If[14 < d ≤ 15, d - 14 - π / 3, -1.0471975511965976], C) → T'''_1
  94. Rotate(T'_3, If[14 < d ≤ 15, ((-(d - 14)) * π / 3), -1.0472], C) → G_1
  95. Rotate(T'', If[14 < d ≤ 15, d - 14 - π / 3, -1.0471975511965976], C) → T'''_2
  96. Segment(G_1, T'''_2) → g_2
  97. Dilate(T, If[15 < d ≤ 16, 16 - d, 0], M) → T'_4
  98. Vector(T, T'_4) → u_1
  99. Translate(T, u_1) → T'_5
  100. Translate(T'_1, u_1) → H_1
  101. Segment(T'_5, H_1) → h_2
  102. Dilate(T, If[16 < d ≤ 17, 17 - d, 0], N) → T'_6
  103. Vector(T, T'_6) → v_1
  104. Translate(T, v_1) → T'_7
  105. Translate(T'_2, v_1) → I_1
  106. Segment(T'_7, I_1) → i_2
  107. Rotate(T'_7, If[17 < d ≤ 18, (((d - 17) * 4) * π / 3), 4.1887902047863905], N) → J_1
  108. Rotate(I_1, If[17 < d ≤ 18, (((-(d - 17)) * 2) * π / 3), -2.0943951023931953], N) → K_1
  109. Segment(J_1, K_1) → j_2
  110. Rotate(J_1, If[18 < d ≤ 19, (((-(d - 18)) * 5) * π / 3), -5.235987755982988], N) → L_1
  111. Rotate(L_1, If[20 < d ≤ 21, ((-(d - 20)) * π / 18), -0.17453292519943295], N) → N_1