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正十边形魔法演示

正十边形魔法演示
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2018-07-05 08:06
多边形角三角函数几何作图
①本范例动态展示正十边形内部多种有趣现象；②涉及核心数学概念包括多边形性质、角度计算、几何变换（旋转、缩放）；③构造要点：运用了Line, Polygon, Segment, LineBisector, Intersect, Rotate, Dilate, Angle, Circle, Ray, CircleSector, Vector, Translate, OrthogonalLine, Point等GeoGebra命令与对象；④适合高中及以上数学学习者和爱好者探索几何奥秘。

GeoGebra 命令及参数

  1. Line((0, 0, 0), (1, 0, 0)) → m_4
  2. Line((0, 0, 0), (0, 1, 0)) → n_4
  3. Line((0, 0, 0), (0, 0, 1)) → p_4
  4. Polygon(A, B, 10) → poly3, r, s_1, C, D, I, J, K, S, T, U, t, i_1, j_1, k_1, l_1, n_1, p_1, r_1
  5. Segment(S, K) → t_1
  6. Segment(K, J) → a_1
  7. Segment(J, I) → b_1
  8. Segment(I, D) → c_1
  9. Segment(D, C) → d_1
  10. Segment(C, B) → e_1
  11. Segment(B, A) → f_2
  12. Segment(A, U) → g_2
  13. Segment(U, T) → h_2
  14. Segment(T, S) → i_2
  15. LineBisector(a_1) → j_2
  16. LineBisector(i_2) → k_2
  17. Intersect(j_2, k_2) → V
  18. Line(T, I) → l_2
  19. Line(A, J) → m_2
  20. Line(C, K) → n_2
  21. Intersect(l_2, m_2) → W
  22. Rotate(poly3, If[10 < b ≤ 11, ((b - 11) * π / 5), π / 5], V) → poly3'
  23. Rotate(T, If[10 < b ≤ 11, ((b - 10) * π / 5), π / 5], V) → T'
  24. Rotate(S, If[10 < b ≤ 11, ((b - 10) * π / 5), π / 5], V) → S'
  25. Segment(T', S') → p_2
  26. Rotate(S, If[10 < b ≤ 11, ((b - 10) * π / 5), π / 5], V) → S'_1
  27. Rotate(K, If[10 < b ≤ 11, ((b - 10) * π / 5), π / 5], V) → K'
  28. Segment(S'_1, K') → q_2
  29. Rotate(K, If[10 < b ≤ 11, ((b - 10) * π / 5), π / 5], V) → K'_1
  30. Rotate(J, If[10 < b ≤ 11, ((b - 10) * π / 5), π / 5], V) → J'
  31. Segment(K'_1, J') → r_2
  32. Rotate(J, If[10 < b ≤ 11, ((b - 10) * π / 5), π / 5], V) → J'_1
  33. Rotate(I, If[10 < b ≤ 11, ((b - 10) * π / 5), π / 5], V) → I'
  34. Segment(J'_1, I') → s_2
  35. Rotate(I, If[10 < b ≤ 11, ((b - 10) * π / 5), π / 5], V) → I'_1
  36. Rotate(D, If[10 < b ≤ 11, ((b - 10) * π / 5), π / 5], V) → D'
  37. Segment(I'_1, D') → t_2
  38. Rotate(D, If[10 < b ≤ 11, ((b - 10) * π / 5), π / 5], V) → D'_1
  39. Rotate(C, If[10 < b ≤ 11, ((b - 10) * π / 5), π / 5], V) → C'
  40. Segment(D'_1, C') → a_2
  41. Rotate(C, If[10 < b ≤ 11, ((b - 10) * π / 5), π / 5], V) → C'_1
  42. Rotate(B, If[10 < b ≤ 11, ((b - 10) * π / 5), π / 5], V) → B'
  43. Segment(C'_1, B') → b_2
  44. Rotate(B, If[10 < b ≤ 11, ((b - 10) * π / 5), π / 5], V) → B'_1
  45. Rotate(A, If[10 < b ≤ 11, ((b - 10) * π / 5), π / 5], V) → A'
  46. Segment(B'_1, A') → c_2
  47. Rotate(A, If[10 < b ≤ 11, ((b - 10) * π / 5), π / 5], V) → A'_1
  48. Rotate(U, If[10 < b ≤ 11, ((b - 10) * π / 5), π / 5], V) → U'
  49. Segment(A'_1, U') → d_2
  50. Rotate(U, If[10 < b ≤ 11, ((b - 10) * π / 5), π / 5], V) → U'_1
  51. Rotate(T, If[10 < b ≤ 11, ((b - 10) * π / 5), π / 5], V) → T'_1
  52. Rotate(C'_1, If[10 < b ≤ 11, ((b - 10) * π / 5), π / 5], V) → Z
  53. Rotate(B', If[10 < b ≤ 11, ((b - 10) * π / 5), π / 5], V) → B''
  54. Rotate(U, If[10 < b ≤ 11, ((b - 10) * π / 5), π / 5], V) → U'_2
  55. Rotate(T, If[10 < b ≤ 11, ((b - 10) * π / 5), π / 5], V) → T'_2
  56. Segment(U'_2, T'_2) → e_2
  57. Dilate(T, If[12 < b ≤ 13, 13 - b, 0], I) → T'_3
  58. Segment(T, T'_3) → g_3
  59. Dilate(A, If[13 < b ≤ 14, 14 - b, 0], W) → A'_2
  60. Segment(A, A'_2) → h_3
  61. Dilate(W, If[14 < b ≤ 15, 15 - b, 0], J) → W'
  62. Segment(W, W') → i_3
  63. Dilate(C, If[15 < b ≤ 16, 16 - b, 0], W) → C'_2
  64. Segment(C, C'_2) → j_3
  65. Dilate(W, If[16 < b ≤ 17, 17 - b, 0], K) → W'_1
  66. Segment(W, W'_1) → k_3
  67. Angle(I, C, W) → η
  68. Circle(C, (0.4 * r_2)) → c
  69. Rotate(C, 36°, W) → C'_3
  70. Angle(C, W, C'_3) → α
  71. Rotate(C, (-(36°)), I) → C'_4
  72. Angle(C'_4, I, C) → β
  73. Ray(W, C'_3) → f
  74. Ray(I, C'_4) → h
  75. Intersect(j_3, h) → H
  76. Line(I, C) → i
  77. Intersect(i, f) → M
  78. Rotate(C, If[17 < b ≤ 18, ((-(b - 17)) * π / 5), ((-π)) / 5], C) → C'_5
  79. Rotate(C'_2, If[17 < b ≤ 18, ((-(b - 17)) * π / 5), ((-π)) / 5], C) → N
  80. Segment(C'_5, N) → j
  81. Intersect(c, j_3, 1) → O
  82. Rotate(O, If[18 < b ≤ 19, ((-(b - 18)) * π / 5), ((-π)) / 5], C) → O'
  83. CircleSector(C, O', O) → d
  84. Dilate(C, If[20 < b ≤ 21, 21 - b, 0], W) → C'_6
  85. Vector(C, C'_6) → u
  86. Translate(d, u) → d'
  87. Dilate(W, If[18 < b ≤ 19, 19 - b, 0], M) → W'_2
  88. Segment(W, W'_2) → k
  89. Line(C, k) → l
  90. Rotate(d', If[20 < b ≤ 21, (((-(b - 20)) * 3) * π / 5), (-3 * π / 5)], C'_6) → d''
  91. Rotate(d'', If[22 < b ≤ 23, ((-(b - 22)) * π / 5), ((-π)) / 5], W) → d'''
  92. Polygon(C, W, I, J, K, S, T, U, A, B) → poly1, c_3, b_3, w, i_4, j_4, k_4, s, t_3, u_1, a_3
  93. Polygon(C, D, I) → t1, i_5, c_4, d_3
  94. Polygon(C, I, W) → t2, w_1, c_5, i_6
  95. Segment(T, W) → m
  96. Rotate(W, If[23 < b ≤ 24, (((-(b - 23)) * 3) * π / 5), (-3 * π / 5)], W) → W'_3
  97. Rotate(W'_1, If[23 < b ≤ 24, (((-(b - 23)) * 3) * π / 5), (-3 * π / 5)], W) → F
  98. Segment(W'_3, F) → n
  99. Rotate(W, If[24 < b ≤ 25, (((b - 24) * 2) * π / 5), (2 * π / 5)], J) → W'_4
  100. Rotate(W', If[24 < b ≤ 25, (((b - 24) * 2) * π / 5), (2 * π / 5)], J) → W''
  101. Segment(W'_4, W'') → p
  102. OrthogonalLine(C, b_2) → q
  103. Circle(C, d_2) → e
  104. Intersect(e, q) → P, Q
  105. Circle(Q, d_2) → f_1
  106. Intersect(f_1, q, 2) → R
  107. Segment(C, D) → g_1
  108. Rotate(C, If[25 < b ≤ 26, ((-(b - 25)) * π / 2), ((-π)) / 2], C) → C'_7
  109. Rotate(D, If[25 < b ≤ 26, ((-(b - 25)) * π / 2), ((-π)) / 2], C) → D'_2
  110. Segment(C'_7, D'_2) → h_1
  111. Angle(Q, C, D) → γ
  112. Ray(Q, R) → m_1
  113. Rotate(C'_7, If[26 < b ≤ 27, ((-(b - 26)) * π), (-π)], Q) → A_1
  114. Rotate(D'_2, If[26 < b ≤ 27, ((-(b - 26)) * π), (-π)], Q) → B_1
  115. Segment(A_1, B_1) → q_1
  116. Line(D, R) → f_3
  117. Line(W, f_3) → l_3
  118. Point(l_3) → C_1
  119. Angle(A, W, C_1) → δ
  120. Dilate(W, If[27 < b ≤ 28, 28 - b, 0], D) → W'_5
  121. Vector(W, W'_5) → v
  122. Translate(A, v) → A'_3
  123. Translate(A'_2, v) → D_1
  124. Segment(A'_3, D_1) → m_3
  125. Rotate(A'_3, If[27 < b ≤ 28, ((b - 27) * δ), δ], D_1) → E_1
  126. Rotate(D_1, If[27 < b ≤ 28, ((b - 27) * δ), δ], D_1) → F_1
  127. Segment(E_1, F_1) → n_3
  128. Polygon(R, D, C) → t3, c_6, r_3, d_4