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正五边形及其相关图形的动态关系

正五边形及其相关图形的动态关系
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2017-06-08 02:03
圆多边形角向量几何作图
①本范例演示了正五边形与其外接圆以及部分其他线段的动态可变关系；②涉及的核心数学概念包括多边形、圆、角度、向量、几何变换等；③作图要点提炼自GeoGebra命令：Rotate（旋转）、Ray（射线）、Point（点）、Segment（线段）、Circle（圆）、Intersect（交点）、Line（直线）、Angle（角度）、Min（最小值）、CircleSector（圆扇形）、Dilate（缩放）、Vector（向量）、Translate（平移）、Polygon（多边形）、LineBisector（角平分线）、OrthogonalLine（垂线）、CircleArc（圆弧）、Midpoint（中点）；④适合高中及以上数学学习者，用于理解复杂几何图形间的相互关系与变换规律。

GeoGebra 命令及参数

  1. Rotate(A, 108°, B) → A'
  2. Ray(B, A') → g
  3. Ray(B, A) → f
  4. Point(f) → C
  5. Segment(C, B) → n
  6. Circle(C, n) → d_1
  7. Rotate(B, (-(108°)), C) → B'
  8. Ray(C, B') → h
  9. Intersect(d_1, h, 1) → G_1
  10. Circle(G_1, n) → e_1
  11. Point(h) → D_1
  12. Rotate(C, (-(108°)), D_1) → C'
  13. Ray(D_1, C') → i
  14. Point(i) → E
  15. Rotate(D_1, (-(108°)), E) → D'
  16. Ray(E, D') → j
  17. Intersect(g, j) → F_1
  18. Segment(F_1, E) → k
  19. Segment(E, D_1) → l
  20. Line(G_1, l) → g_1
  21. Intersect(e_1, g_1, 1) → J
  22. Line(J, k) → h_1
  23. Intersect(h_1, g) → K
  24. Angle(A, B, A') → α
  25. Angle(B', C, B) → β
  26. Angle(C', D_1, C) → γ
  27. Angle(D', E, D_1) → δ
  28. Segment(D_1, C) → m
  29. Segment(B, F_1) → p
  30. Min(list) → a
  31. Circle(B, (0.2 * n)) → q
  32. Intersect(q, p) → H,
  33. Intersect(q, n) → I,
  34. CircleSector(B, I, H) → r
  35. Dilate(B, If[0 < o ≤ 1, 1 - o, 0], C) → B'_1
  36. Vector(B, B'_1) → u
  37. Translate(r, u) → r'
  38. Rotate(r', If[1 < o ≤ 2, (((o - 1) * 2) * π / 5), (2 * π / 5)], C) → r''
  39. Dilate(C, If[2 < o ≤ 3, 3 - o, 0], G_1) → C'_1
  40. Vector(C, C'_1) → v
  41. Translate(r'', v) → r'''
  42. Rotate(r''', If[3 < o ≤ 4, (((o - 3) * 2) * π / 5), (2 * π / 5)], G_1) → s
  43. Dilate(G_1, If[4 < o ≤ 5, 5 - o, 0], J) → D'_1
  44. Vector(G_1, D'_1) → w
  45. Translate(s, w) → s'
  46. Rotate(s', If[5 < o ≤ 6, (((o - 5) * 2) * π / 5), (2 * π / 5)], J) → s''
  47. Dilate(J, If[6 < o ≤ 7, 7 - o, 0], K) → E'
  48. Vector(J, E') → t
  49. Translate(s'', t) → s'''
  50. Circle(J, n) → k_1
  51. Rotate(s''', If[7 < o ≤ 8, (((o - 7) * 2) * π / 5), (2 * π / 5)], K) → c_1
  52. Segment(B, B'_1) → b
  53. Segment(C, C'_1) → c
  54. Segment(G_1, D'_1) → d
  55. Segment(J, E') → e
  56. Dilate(K, If[8 < o ≤ 9, 9 - o, 0], B) → K'
  57. Segment(K, K') → f_1
  58. Polygon(B, C, G_1, J, K) → poly1, b_1, c_2, g_2, j_1, k_2
  59. Rotate(B, If[9 < o ≤ 10, ((-(o - 9)) * α), (-α)], C) → B'_2
  60. Rotate(B'_1, If[9 < o ≤ 10, ((-(o - 9)) * α), (-α)], C) → L
  61. Segment(B'_2, L) → b'
  62. Rotate(B'_2, If[10 < o ≤ 11, ((-(o - 10)) * α), (-α)], G_1) → M
  63. Rotate(L, If[10 < o ≤ 11, ((-(o - 10)) * α), (-α)], G_1) → L'
  64. Segment(M, L') → b''
  65. Rotate(M, If[11 < o ≤ 12, ((-(o - 11)) * α), (-α)], J) → M'
  66. Rotate(L', If[12 < o ≤ 13, ((-(o - 12)) * α), (-α)], J) → L''
  67. Segment(M', L'') → b'''
  68. Rotate(M', If[12 < o ≤ 13, ((-(o - 12)) * α), (-α)], K) → M''
  69. Rotate(L'', If[12 < o ≤ 13, ((-(o - 12)) * α), (-α)], K) → L'''
  70. Segment(M'', L''') → i_1
  71. LineBisector(B, C) → l_1
  72. LineBisector(G_1, J) → m_1
  73. Intersect(m_1, l_1) → N
  74. Circle(N, J) → p_1
  75. Intersect(p_1, m_1) → D, P
  76. Line(N, G_1) → n_1
  77. Line(J, G_1) → q_1
  78. Intersect(q_1, m_1) → F
  79. OrthogonalLine(F, n_1) → r_1
  80. Intersect(r_1, n_1) → G
  81. Line(J, D) → s_1
  82. Intersect(n_1, s_1) → O
  83. Angle(F, G, N) → ε
  84. Line(N, G_1) → t_1
  85. Intersect(p_1, t_1) → Q, R
  86. Rotate(J, If[13 < o ≤ 14, (((o - 13) * 288) * π / 180), (288 * π / 180)], N) → J'
  87. CircleArc(N, J, J') → d_2
  88. Rotate(G_1, If[14 < o ≤ 15, (((o - 14) * 36) * π / 180), (36 * π / 180)], N) → S
  89. CircleArc(N, G_1, S) → e_2
  90. Rotate(D, If[15 < o ≤ 16, (((o - 15) * 36) * π / 180), (36 * π / 180)], N) → D'_2
  91. CircleArc(N, D, D'_2) → f_2
  92. CircleArc(N, J, K) → h_2
  93. CircleArc(N, K, B) → p_2
  94. CircleArc(N, B, C) → q_2
  95. CircleArc(N, C, G_1) → r_2
  96. CircleArc(N, G_1, J) → s_2
  97. Rotate(e_2, If[18 < o ≤ 19, (((o - 18) * 36) * π / 180), (36 * π / 180)], N) → t_2
  98. Dilate(N, If[19 < o ≤ 20, 20 - o, 0], F) → N'
  99. Segment(N, N') → a_1
  100. Dilate(F, If[20 < o ≤ 21, 21 - o, 0], D) → F'
  101. Segment(F, F') → i_2
  102. Dilate(F, If[21 < o ≤ 22, 22 - o, 0], G) → F'_1
  103. Dilate(N, If[21 < o ≤ 22, 22 - o, 0], G) → N'_1
  104. Segment(N, N'_1) → j_2
  105. Segment(F, F'_1) → l_2
  106. Dilate(G, If[22 < o ≤ 23, 23 - o, 0], O) → G'
  107. Dilate(J, If[22 < o ≤ 23, 23 - o, 0], O) → J'_1
  108. Segment(J, J'_1) → m_2
  109. Segment(G, G') → n_2
  110. Dilate(N, If[22 < o ≤ 23, 23 - o, 0], Q) → N'_2
  111. Segment(N, N'_2) → a_2
  112. Midpoint(G, N) → T
  113. Rotate(G, If[24.5 < o ≤ 25.5, ((-(o - 24.5)) * π), (-π)], T) → G'_1
  114. Rotate(G', If[24.5 < o ≤ 25.5, ((-(o - 24.5)) * π), (-π)], T) → G''
  115. Segment(G'_1, G'') → b_2