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六边形内共点关系动态演示

六边形内共点关系动态演示
几何画板及绘图内容加载中，请稍等……
2017-05-16 02:23
多边形向量几何作图
①本范例动态演示任意六边形内共点关系的几何构造；②涉及核心数学概念包括多边形、向量、几何变换（旋转、平移等）；③作图要点：运用了中点、多边形构造、缩放、线段、旋转、交点等 GeoGebra 命令与对象；④适合高中及以上数学教学或几何探究，可用于理解复杂几何图形的性质。

GeoGebra 命令及参数

  1. Midpoint(D, E) → B_2
  2. Midpoint(F, A) → V_1
  3. Midpoint(B_2, V_1) → M
  4. Midpoint(A, B) → W_1
  5. Midpoint(E, F) → U_1
  6. Midpoint(C, D) → A_2
  7. Midpoint(B, C) → Z_1
  8. Polygon(A, B, C, D, E, F) → poly1, a, b, c, d, e, f
  9. Dilate(A, If[0 < o ≤ 1, 1 - o, 0], W_1) → A'
  10. Segment(A, A') → k_1
  11. Dilate(B, If[0 < o ≤ 1, 1 - o, 0], Z_1) → B'
  12. Segment(B, B') → l_1
  13. Dilate(C, If[0 < o ≤ 1, 1 - o, 0], A_2) → C'
  14. Segment(C, C') → m_1
  15. Dilate(D, If[0 < o ≤ 1, 1 - o, 0], B_2) → D'
  16. Segment(D, D') → n_1
  17. Dilate(E, If[0 < o ≤ 1, 1 - o, 0], U_1) → E'
  18. Segment(E, E') → p_1
  19. Dilate(F, If[0 < o ≤ 1, 1 - o, 0], V_1) → F'
  20. Segment(F, F') → q_1
  21. Rotate(C, If[1.5 < o ≤ 2.5, ((-(o - 1.5)) * π / 3), ((-π)) / 3], C) → C'_1
  22. Rotate(C', If[1.5 < o ≤ 2.5, ((-(o - 1.5)) * π / 3), ((-π)) / 3], C) → C''
  23. Rotate(D, If[1.5 < o ≤ 2.5, ((-(o - 1.5)) * π / 3), ((-π)) / 3], D) → D'_1
  24. Rotate(D', If[1.5 < o ≤ 2.5, ((-(o - 1.5)) * π / 3), ((-π)) / 3], D) → D''
  25. Rotate(E, If[1.5 < o ≤ 2.5, ((-(o - 1.5)) * π / 3), ((-π)) / 3], E) → E'_1
  26. Rotate(E', If[1.5 < o ≤ 2.5, ((-(o - 1.5)) * π / 3), ((-π)) / 3], E) → E''
  27. Rotate(F, If[1.5 < o ≤ 2.5, ((-(o - 1.5)) * π / 3), ((-π)) / 3], F) → F'_1
  28. Rotate(F', If[1.5 < o ≤ 2.5, ((-(o - 1.5)) * π / 3), ((-π)) / 3], F) → F''
  29. Rotate(A, If[1.5 < o ≤ 2.5, ((-(o - 1.5)) * π / 3), ((-π)) / 3], A) → A'_1
  30. Rotate(A', If[1.5 < o ≤ 2.5, ((-(o - 1.5)) * π / 3), ((-π)) / 3], A) → A''
  31. Rotate(B, If[1.5 < o ≤ 2.5, ((-(o - 1.5)) * π / 3), ((-π)) / 3], B) → B'_1
  32. Rotate(B', If[1.5 < o ≤ 2.5, ((-(o - 1.5)) * π / 3), ((-π)) / 3], B) → B''
  33. Rotate(C, If[2.5 < o ≤ 3.5, ((o - 2.5) * π / 3), π / 3], D) → C'_2
  34. Rotate(C', If[2.5 < o ≤ 3.5, ((o - 2.5) * π / 3), π / 3], D) → C''_1
  35. Rotate(D, If[2.5 < o ≤ 3.5, ((o - 2.5) * π / 3), π / 3], E) → D'_2
  36. Rotate(D', If[2.5 < o ≤ 3.5, ((o - 2.5) * π / 3), π / 3], E) → D''_1
  37. Rotate(E, If[2.5 < o ≤ 3.5, ((o - 2.5) * π / 3), π / 3], F) → E'_2
  38. Rotate(E', If[2.5 < o ≤ 3.5, ((o - 2.5) * π / 3), π / 3], F) → E''_1
  39. Rotate(F, If[2.5 < o ≤ 3.5, ((o - 2.5) * π / 3), π / 3], A) → F'_2
  40. Rotate(F', If[2.5 < o ≤ 3.5, ((o - 2.5) * π / 3), π / 3], A) → F''_1
  41. Rotate(A, If[2.5 < o ≤ 3.5, ((o - 2.5) * π / 3), π / 3], B) → A'_2
  42. Rotate(A', If[2.5 < o ≤ 3.5, ((o - 2.5) * π / 3), π / 3], B) → A''_1
  43. Rotate(B, If[2.5 < o ≤ 3.5, ((o - 2.5) * π / 3), π / 3], C) → B'_2
  44. Rotate(B', If[2.5 < o ≤ 3.5, ((o - 2.5) * π / 3), π / 3], C) → B''_1
  45. Rotate(A, If[1 < o ≤ 2, ((-(o - 1)) * π), (-π)], W_1) → A'_6
  46. Rotate(A', If[1 < o ≤ 2, ((-(o - 1)) * π), (-π)], W_1) → A''_2
  47. Segment(A'_6, A''_2) → r_1
  48. Rotate(B, If[1 < o ≤ 2, ((-(o - 1)) * π), (-π)], Z_1) → B'_6
  49. Rotate(B', If[1 < o ≤ 2, ((-(o - 1)) * π), (-π)], Z_1) → B''_2
  50. Segment(B'_6, B''_2) → s_1
  51. Rotate(C, If[1 < o ≤ 2, ((-(o - 1)) * π), (-π)], A_2) → C'_7
  52. Rotate(C', If[1 < o ≤ 2, ((-(o - 1)) * π), (-π)], A_2) → C''_2
  53. Segment(C'_7, C''_2) → t_1
  54. Rotate(D, If[1 < o ≤ 2, ((-(o - 1)) * π), (-π)], B_2) → D'_5
  55. Rotate(D', If[1 < o ≤ 2, ((-(o - 1)) * π), (-π)], B_2) → D''_2
  56. Segment(D'_5, D''_2) → a_1
  57. Rotate(E, If[1 < o ≤ 2, ((-(o - 1)) * π), (-π)], U_1) → E'_6
  58. Rotate(E', If[1 < o ≤ 2, ((-(o - 1)) * π), (-π)], U_1) → E''_2
  59. Segment(E'_6, E''_2) → b_1
  60. Rotate(F, If[1 < o ≤ 2, ((-(o - 1)) * π), (-π)], V_1) → F'_6
  61. Rotate(F', If[1 < o ≤ 2, ((-(o - 1)) * π), (-π)], V_1) → F''_2
  62. Segment(F'_6, F''_2) → c_1
  63. Midpoint(Z_1, B_2) → E_2
  64. Dilate(Z_1, If[2 < o ≤ 3, 3 - o, 0], E_2) → G
  65. Segment(Z_1, G) → g
  66. Dilate(B_2, If[2 < o ≤ 3, 3 - o, 0], M) → H
  67. Segment(B_2, H) → h
  68. Rotate(Z_1, If[3 < o ≤ 4, ((-(o - 3)) * π), (-π)], E_2) → I
  69. Rotate(G, If[3 < o ≤ 4, ((-(o - 3)) * π), (-π)], E_2) → G'
  70. Segment(I, G') → g'
  71. Rotate(B_2, If[3 < o ≤ 4, ((-(o - 3)) * π), (-π)], M) → N
  72. Rotate(H, If[3 < o ≤ 4, ((-(o - 3)) * π), (-π)], M) → H'_1
  73. Segment(N, H'_1) → h'
  74. Midpoint(V_1, Z_1) → J
  75. Dilate(V_1, If[2 < o ≤ 3, 3 - o, 0], J) → K
  76. Segment(V_1, K) → i
  77. Rotate(V_1, If[3 < o ≤ 4, ((-(o - 3)) * π), (-π)], J) → L
  78. Rotate(K, If[3 < o ≤ 4, ((-(o - 3)) * π), (-π)], J) → K'
  79. Segment(L, K') → i'
  80. Polygon(V_1, Z_1, B_2) → poly2, b_2, v_1, z_1
  81. Midpoint(W_1, A_2) → O
  82. Midpoint(A_2, U_1) → P
  83. Midpoint(U_1, W_1) → Q
  84. Dilate(W_1, If[4 < o ≤ 5, 5 - o, 0], O) → R
  85. Dilate(A_2, If[4 < o ≤ 5, 5 - o, 0], P) → S
  86. Dilate(U_1, If[4 < o ≤ 5, 5 - o, 0], Q) → T
  87. Segment(W_1, R) → j
  88. Segment(A_2, S) → k
  89. Segment(U_1, T) → l
  90. Rotate(A_2, If[5 < o ≤ 6, ((-(o - 5)) * π), (-π)], P) → U
  91. Rotate(S, If[5 < o ≤ 6, ((-(o - 5)) * π), (-π)], P) → S'
  92. Segment(U, S') → k'
  93. Rotate(W_1, If[5 < o ≤ 6, ((-(o - 5)) * π), (-π)], O) → V
  94. Rotate(R, If[5 < o ≤ 6, ((-(o - 5)) * π), (-π)], O) → R'
  95. Segment(V, R') → j'
  96. Rotate(U_1, If[5 < o ≤ 6, ((-(o - 5)) * π), (-π)], Q) → W
  97. Rotate(T, If[5 < o ≤ 6, ((-(o - 5)) * π), (-π)], Q) → T'
  98. Segment(W, T') → l'
  99. Polygon(W_1, A_2, U_1) → poly3, u_1, w_1, a_2
  100. Segment(U_1, O) → m
  101. Segment(W_1, P) → n
  102. Intersect(m, n) → Z
  103. Dilate(B_2, If[6 < o ≤ 7, 7 - o, 0], J) → A_1
  104. Segment(B_2, A_1) → p
  105. Dilate(V_1, If[7 < o ≤ 8, 8 - o, 0], Z) → B_1
  106. Dilate(Z, If[8 < o ≤ 9, 9 - o, 0], Q) → Z'
  107. Segment(V_1, B_1) → q
  108. Dilate(Z, If[8 < o ≤ 9, 9 - o, 0], E_2) → Z'_1
  109. Segment(Z, Z'_1) → r
  110. Dilate(U_1, If[9 < o ≤ 10, 10 - o, 0], Z) → C_1
  111. Segment(U_1, C_1) → s
  112. Dilate(Z, If[10 < o ≤ 11, 11 - o, 0], O) → Z'_2
  113. Segment(ZZ, Z'_2) → t
  114. Dilate(W_1, If[11 < o ≤ 12, 12 - o, 0], Z) → D_1
  115. Segment(W_1, D_1) → f_1
  116. Dilate(Z, If[12 < o ≤ 13, 13 - o, 0], P) → Z'_3
  117. Segment(Z, Z'_3) → g_1