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五边形与三角形构成的共圆点动态演示

五边形与三角形构成的共圆点动态演示
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2017-04-24 10:12
圆多边形向量三角函数几何作图
①本范例动态演示了两个正五边形与一个等边三角形如何构成四个共圆点；②涉及的核心数学概念包括正多边形、共圆点、旋转、相似、向量等；③作图要点提炼自构造摘要，使用了Rotate（旋转）、Circle（圆）、Intersect（交点）、Vector（向量）等GeoGebra命令与对象；④适合高中及以上数学学习者，用于理解复杂几何图形的性质与构造方法。

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
  10. Circle(G, n) → e_1
  11. Point(h) → D
  12. Rotate(C, (-(108°)), D) → C'
  13. Ray(D, C') → i
  14. Point(i) → E
  15. Rotate(D, (-(108°)), E) → D'
  16. Ray(E, D') → j
  17. Intersect(g, j) → F
  18. Segment(F, E) → k
  19. Segment(E, D) → l
  20. Line(G, 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, C) → γ
  27. Angle(D', E, D) → δ
  28. Segment(D, C) → m
  29. Segment(B, F) → 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) → 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) → s
  43. Dilate(G, If[4 < o ≤ 5, 5 - o, 0], J) → D'_1
  44. Vector(G, 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, 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, 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) → M
  63. Rotate(L, If[10 < o ≤ 11, ((-(o - 10)) * α), (-α)], G) → 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. Rotate(B, If[13 < o ≤ 14, ((-(o - 13)) * π / 3), ((-π)) / 3], B) → B'_3
  72. Rotate(B'_1, If[13 < o ≤ 14, ((-(o - 13)) * π / 3), ((-π)) / 3], B) → N
  73. Segment(B'_3, N) → b'_1
  74. Rotate(C, ((-π)) / 3, B) → C'_2
  75. Segment(J, C'_2) → l_1
  76. Rotate(B, If[14 < o ≤ 15, ((o - 14) * π / 3), π / 3], C) → B'_4
  77. Rotate(B'_1, If[14 < o ≤ 15, ((o - 14) * π / 3), π / 3], C) → O
  78. Segment(B'_4, O) → b'_2
  79. Dilate(J, If[15 < o ≤ 16, 16 - o, 0], C'_2) → J'
  80. Segment(J, J') → m_1
  81. Polygon(B, C, C'_2) → poly2, c'_2, b_2, c_3
  82. Dilate(B, If[17 < o ≤ 18, 18 - o, 0], C'_2) → B'_5
  83. Vector(B, B'_5) → u_1
  84. Translate(B, u_1) → B'_6
  85. Translate(C, u_1) → C'_3
  86. Translate(G, u_1) → G'
  87. Translate(J, u_1) → J'_1
  88. Translate(K, u_1) → K'_1
  89. Polygon(B'_6, C'_3, G', J'_1, K'_1) → poly1', b'_6, c'_3, g', j'_1, k'_1
  90. Rotate(B'_6, If[18 < o ≤ 19, ((o - 18) * π / 2), π / 2], C'_2) → P
  91. Rotate(C'_3, If[18 < o ≤ 19, ((o - 18) * π / 2), π / 2], C'_2) → Q
  92. Rotate(G', If[18 < o ≤ 19, ((o - 18) * π / 2), π / 2], C'_2) → G''
  93. Rotate(J'_1, If[18 < o ≤ 19, ((o - 18) * π / 2), π / 2], C'_2) → R
  94. Rotate(K'_1, If[18 < o ≤ 19, ((o - 18) * π / 2), π / 2], C'_2) → S
  95. Polygon(P, Q, G'', R, S) → poly1'', p_1, q_1, g'', r_1, s_1
  96. Segment(C'_2, J) → n_1
  97. Distance(C'_2, b) → a_1
  98. Circle(C'_2, a_1) → t_1
  99. Distance(C'_2, J) → o_1
  100. Circle(C'_2, b) → d_2
  101. Intersect(m_1, n_1) → T
  102. Intersect(d_2, n_1) → U,
  103. Dilate(U, If[19.5 < o ≤ 20.5, 20.5 - o, 0], J) → U'
  104. Polygon(C'_2, U', 5) → poly3, f_2, h_2, i_2, j_2, l_2, V, W, Z
  105. LineBisector(W, Z) → m_2
  106. LineBisector(E', W) → n_2
  107. Intersect(n_2, m_2) → A_1
  108. Circle(A_1, W) → e_2
  109. Dilate(A_1, If[20.5 < o ≤ 21.5, 21.5 - o, 0], W) → B_1
  110. Dilate(A_1, If[21 < o ≤ 22, 22 - o, 0], W) → C_1
  111. Segment(A_1, C_1) → p_2
  112. Angle(E', A_1, WW) → ε
  113. Angle(B, A_1, E') → ζ
  114. Angle(Z, A_1, B) → μ
  115. Angle(WW, A_1, Z) → θ
  116. Rotate(WW, If[22 < o ≤ 23, ((-(o - 22)) * ε), (-ε)], A_1) → WW'
  117. CircleArc(A_1, WW', WW) → q_2
  118. Rotate(E', If[23 < o ≤ 24, ((-(o - 23)) * ζ), (-ζ)], A_1) → E''
  119. CircleArc(A_1, E'', E') → r_2
  120. Rotate(B, If[24 < o ≤ 25, ((-(o - 24)) * μ), (-μ)], A_1) → B'_7
  121. CircleArc(A_1, B'_7, B) → s_2
  122. Rotate(Z, If[25 < o ≤ 26, ((-(o - 25)) * θ), (-θ)], A_1) → Z'
  123. CircleArc(A_1, Z', Z) → t_2

AI 绘图提示语参考

画两个正五边形和一个等边三角形，展示它们如何形成四个共圆点。首先旋转其中一个五边形顶点108度，然后从旋转中心向两个顶点引射线，交点与顶点连线形成圆。再旋转五边形和三角形顶点，重复构造圆和交点，最终得到四个共圆点。用向量表示旋转，用圆和交点强调共圆关系，整体动态展示几何性质。
以上提示语为根据原始绘图命令整理而来，仅供参考。