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平行四边形中的奇妙现象

平行四边形中的奇妙现象
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2017-09-07 11:31
多边形变换几何作图
本范例展示了当在平行四边形的两边上添加平行线段时，所发生的动态且可修改的关系。核心数学概念包括平行线、等边三角形和整数。通过使用GeoGebra命令如Line、Rotate、Segment等进行构造。适合初中及以上学生学习几何作图和变换。

GeoGebra 命令及参数

  1. Line(A, B) → r_2
  2. Rotate(B, a°, A) → B'
  3. Ray(A, B') → g
  4. Line(B, g) → h
  5. Point(g) → C
  6. Segment(A, C) → p_1
  7. Point(p_1) → T
  8. Ray(A, B) → f
  9. Line(C, f) → i
  10. Intersect(h, i) → D
  11. Segment(C, D) → q_1
  12. Line(T, q_1) → t_1
  13. Point(q_1) → U
  14. Line(U, p_1) → s_1
  15. Intersect(t_1, s_1) → Z
  16. Line(C, Z) → g_2
  17. Segment(T, C) → j_1
  18. Segment(A, T) → i_1
  19. Segment(U, D) → l_1
  20. Segment(C, U) → k_1
  21. Angle(B, A, B') → α
  22. Segment(C, D) → j
  23. Segment(A, B) → k
  24. Segment(B, D) → l
  25. Segment(A, C) → m
  26. Min({i_1, j_1, k_1, l_1}) → b
  27. Circle(C, (0.3 * b)) → d
  28. Circle(A, (0.35 * b)) → e
  29. Circle(B, (0.4 * b)) → p
  30. Circle(D, (0.3 * b)) → c
  31. Intersect(e, k, 1) → E
  32. Intersect(e, m, 1) → F
  33. Intersect(p, k, 1) → G
  34. Intersect(p, l, 1) → H
  35. Intersect(d, g, 2) → I
  36. Ray(C, I) → q
  37. Intersect(c, h, 2) → J
  38. Ray(D, J) → r
  39. Intersect(c, i, 2) → K
  40. Ray(D, K) → s
  41. Dilate(A, If[0 < o ≤ 1, 1 - o, 0], B) → A'
  42. Segment(A, A') → t
  43. Rotate(E, If[1 < o ≤ 2, ((o - 1) * α), α], A) → E'
  44. CircleSector(A, E, E') → c_1
  45. Dilate(A, If[2 < o ≤ 3, 3 - o, 0], C) → A'_1
  46. Vector(A, A'_1) → u
  47. Translate(c_1, u) → d_1
  48. Segment(A, A'_1) → f_1
  49. Dilate(C, If[3 < o ≤ 4, 4 - o, 0], D) → C'
  50. Vector(C, C') → v
  51. Translate(d_1, v) → e_1
  52. Segment(C, C') → g_1
  53. Dilate(D, If[4 < o ≤ 5, 5 - o, 0], B) → D'
  54. Segment(D, D') → h_1
  55. Polygon(A, B, D, C) → poly1, a_1, b_1, d_2, c_2
  56. Dilate(A, If[5.5 < o ≤ 6.5, 6.5 - o, 0], C) → A'_2
  57. Vector(A, A'_2) → w
  58. Translate(A, w) → A'_3
  59. Translate(A', w) → A''
  60. Dilate(A, If[6.5 < o ≤ 7.5, 7.5 - o, 0], B) → A'_4
  61. Vector(A, A'_4) → u_1
  62. Translate(A, u_1) → A'_5
  63. Translate(A'_1, u_1) → L
  64. Intersect(d, i) → M, N
  65. Intersect(d, g) → O, P
  66. Angle(A, C, N) → β
  67. Rotate(N, If[6.5 < o ≤ 7.5, ((-(o - 6.5)) * β), (-β)], C) → N'
  68. Dilate(C, If[7.5 < o ≤ 8.5, 8.5 - o, 0], B) → C'_1
  69. Vector(C, C'_1) → v_1
  70. Segment(D, B) → r_1
  71. Segment(A, B) → f_2
  72. Intersect(f_2, s_1) → V
  73. Intersect(t_1, r_1) → W
  74. Segment(D, V) → i_2
  75. Segment(A, W) → j_2
  76. Intersect(g_2, r_2) → A_1
  77. Dilate(T, If[6.5 < o ≤ 7.5, 7.5 - o, 0], W) → T'
  78. Segment(T, T') → m_1
  79. Dilate(U, If[7.5 < o ≤ 8.5, 8.5 - o, 0], Z) → U'
  80. Segment(U, U') → n_1
  81. Dilate(Z, If[8.5 < o ≤ 9.5, 9.5 - o, 0], V) → Z'
  82. Segment(Z, Z') → h_2
  83. Dilate(A, If[9.5 < o ≤ 10.5, 10.5 - o, 0], T) → A'_6
  84. Vector(A, A'_6) → w_1
  85. Translate(c_1, w_1) → e_2
  86. Dilate(D, If[10.5 < o ≤ 11.5, 11.5 - o, 0], U) → C'_2
  87. Vector(D, C'_2) → u_2
  88. Dilate(T, If[10.5 < o ≤ 11.5, 11.5 - o, 0], Z) → T'_1
  89. Vector(T, T'_1) → v_2
  90. Translate(e_2, v_2) → k_2
  91. Intersect(g_2, j_2) → Q
  92. Dilate(C, If[11.5 < o ≤ 12.5, 12.5 - o, 0], A_1) → C'_3
  93. Segment(C, C'_3) → l_2
  94. Dilate(A, If[12.5 < o ≤ 13.5, 13.5 - o, 0], Q) → A'_7
  95. Segment(A, A'_7) → m_2
  96. Dilate(Q, If[13.5 < o ≤ 14.5, 14.5 - o, 0], W) → Q'
  97. Segment(Q, Q') → n_2
  98. Dilate(D, If[14.5 < o ≤ 15.5, 15.5 - o, 0], Q) → D'_1
  99. Segment(D, D'_1) → p_2
  100. Intersect(l_2, m_2) → R
  101. Dilate(Q, If[15.5 < o ≤ 16.5, 16.5 - o, 0], V) → Q'_1
  102. Segment(Q, Q'_1) → q_2