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Cut-The-Knot 动作 19：动态四边形与二次方程的可视化探索

Cut-The-Knot 动作 19：动态四边形与二次方程的可视化探索
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2018-02-17 12:08
多边形角几何作图滑块动画
① 本范例是一个动态且可修改的几何演示，源自 Alexander Bogomolny 在 Twitter 上发布的一道数学问题，通过交互操作展示一般四边形（含风筝形）中隐藏的几何关系。② 核心数学概念涉及一般四边形、风筝形、二次方程、标准差、割线以及角度与线段之间的约束关系，体现了代数与几何的交叉。③ 作图要点：从构造摘要可见，使用了 Segment、Rotate、Ray、Point、Line、Angle、Intersect、OrthogonalLine、Polygon、Min、Circle、CircleSector、Dilate、Vector、Translate 等 GeoGebra 命令，构建了多个点（如 A、B、C、D、E、F、G）、线段、圆、扇形、向量及平移对象，并利用滑块或动态点实现实时变化。④ 适合高中及以上学段的学生或数学爱好者，用于探究四边形性质、二次方程与几何约束的关联，也可作为教师课堂演示或学生自主探索的素材。

GeoGebra 命令及参数

  1. Segment(A, B) → f
  2. Rotate(A, (a * π / 180), B) → A'
  3. Ray(B, A') → g
  4. Point(g) → C
  5. Line(C, A) → n_2
  6. Segment(C, A) → i
  7. Angle(B, C, A) → γ
  8. Rotate(B, γ / 2, C) → B'
  9. Ray(C, B') → k
  10. Intersect(f, k) → I
  11. Angle(C, A, B) → β
  12. Rotate(C, β / 2, A) → C'
  13. Ray(A, C') → j
  14. Segment(B, C) → h
  15. Intersect(h, j) → J
  16. Segment(J, I) → l
  17. Point(l) → K
  18. OrthogonalLine(K, i) → h_1
  19. Line(C, B) → m_2
  20. OrthogonalLine(K, h) → g_2
  21. Line(B, A) → l_2
  22. OrthogonalLine(K, f) → n
  23. Angle(A, B, A') → α
  24. Polygon(B, A, C) → t1, c, b, a_1
  25. Angle(C, A, C') → δ
  26. Angle(B, C, B') → ε
  27. Intersect(j, k) → D
  28. Min({f, i, h}) → d
  29. Circle(B, (0.15 * d)) → e
  30. Circle(A, (0.15 * d)) → p
  31. Circle(C, (0.15 * d)) → q
  32. Intersect(q, h, 1) → E
  33. Intersect(p, i) → F,
  34. Intersect(e, f) → G,
  35. Intersect(e, h, 1) → H
  36. CircleSector(B, G, H) → r
  37. OrthogonalLine(K, l) → m
  38. OrthogonalLine(K, h) → s
  39. Intersect(s, h) → N
  40. Intersect(n, l_2) → M
  41. Segment(K, M) → t
  42. Segment(K, N) → f_1
  43. Angle(K, N, C) → ζ
  44. Angle(A, M, K) → η
  45. OrthogonalLine(D, i) → g_1
  46. Circle(K, f_1) → c_1
  47. Intersect(n_2, h_1) → L
  48. Angle(K, L, A) → θ
  49. Ray(K, L) → i_1
  50. Circle(K, f_1) → d_1
  51. Intersect(d_1, i_1, 1) → O
  52. Dilate(A, If[0 < u ≤ 1, 1 - u, 0], D) → A'_1
  53. Dilate(C, If[0 < u ≤ 1, 1 - u, 0], D) → C'_1
  54. Segment(A, A'_1) → j_1
  55. Segment(C, C'_1) → k_1
  56. Dilate(D, If[1 < u ≤ 2, 2 - u, 0], J) → D'
  57. Dilate(D, If[1 < u ≤ 2, 2 - u, 0], I) → D'_1
  58. Segment(D, D') → l_1
  59. Segment(D, D'_1) → m_1
  60. Rotate(F, If[1 < u ≤ 2, ((u - 1) * δ), δ], A) → F'
  61. Rotate(E, If[1 < u ≤ 2, ((u - 1) * ε), ε], C) → E'
  62. CircleSector(A, F, F') → e_1
  63. CircleSector(C, E, E') → p_1
  64. Rotate(p_1, If[3 < u ≤ 4, ((u - 3) * ε), ε], C) → q_1
  65. Rotate(e_1, If[3 < u ≤ 4, ((u - 3) * δ), δ], A) → r_1
  66. Dilate(J, If[4 < u ≤ 5, 5 - u, 0], I) → J'
  67. Segment(J, J') → n_1
  68. Dilate(K, If[7 < u ≤ 8, 8 - u, 0], M) → K'
  69. Dilate(K, If[7 < u ≤ 8, 8 - u, 0], N) → K'_1
  70. Segment(K, K') → s_1
  71. Segment(K, K'_1) → t_1
  72. Dilate(K, If[7 < u ≤ 8, 8 - u, 0], O) → K'_2
  73. Segment(K, K'_2) → b_1
  74. Dilate(O, If[8 < u ≤ 9, 9 - u, 0], L) → O'
  75. Segment(O, O') → f_2
  76. Intersect(g_2, m_2) → P
  77. Angle(L, K, P) → κ
  78. Rotate(K, If[10 < u ≤ 11, ((-(u - 10)) * κ), (-κ)], K) → K'_3
  79. Rotate(K'_1, If[10 < u ≤ 11, ((-(u - 10)) * κ), (-κ)], K) → Q
  80. Segment(K'_3, Q) → h_2
  81. Line(O, s_1) → i_2
  82. Point(i_2) → R
  83. Angle(R, O, L) → λ
  84. Dilate(K, If[11 < u ≤ 12, 12 - u, 0], O) → K'_4
  85. Vector(K, K'_4) → v
  86. Translate(K, v) → K'_5
  87. Translate(K', v) → K''
  88. Segment(K'_5, K'') → j_2
  89. Rotate(K'_5, If[12 < u ≤ 13, ((u - 12) * λ), λ], O) → S
  90. Rotate(K'', If[12 < u ≤ 13, ((u - 12) * λ), λ], O) → K'''
  91. Segment(S, K''') → k_2