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SolveMyMaths 行动1：动态可调的面积与向量几何问题

SolveMyMaths 行动1：动态可调的面积与向量几何问题
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2017-11-25 09:30
多边形向量几何作图滑块动画
① 本范例是一个动态且可修改的几何演示，重现了 Ed Southall 在 Twitter 上提出的一个数学问题，通过滑块和拖拽点直观展示图形变化。② 核心数学概念包括分数、逻辑推理、加法、面积计算以及向量运算，涉及多边形、圆、扇形和旋转等几何元素。③ 作图要点：使用 GeoGebra 的 Segment、Rotate、Angle、Ray、Point、Polygon、Circle、Intersect、CircleSector、Dilate、Vector、Translate、Min、Line、Midpoint 等命令，构建了多个点、线段、圆、扇形和向量对象，并通过动态参数控制图形状态。④ 适合中学及以上学段的学生和教师，用于探索几何面积关系、向量分解和动态数学建模，也可作为数学竞赛或课堂讨论的素材。

GeoGebra 命令及参数

  1. Segment(A, B) → f
  2. Rotate(B, (a * π / 180), A) → B'
  3. Angle(B, A, B') → α
  4. Ray(A, B') → g
  5. Point(g) → C
  6. Segment(A, C) → h
  7. Segment(C, B) → i
  8. Polygon(A, B, C) → t1, c, a_1, b
  9. Circle(A, (0.15 * Min[h, f])) → d
  10. Intersect(d, f) → D,
  11. Intersect(d, h) → E,
  12. CircleSector(A, D, E) → e
  13. Rotate(B, π / 3, C) → B'_1
  14. Rotate(C, π / 3, A) → C'
  15. Rotate(B, π / 3, A) → B'_2
  16. Rotate(A, If[0 < j ≤ 1, (j * π / 3), π / 3], A) → A'
  17. Rotate(C, If[0 < j ≤ 1, (j * π / 3), π / 3], A) → C'_1
  18. Segment(A', C'_1) → h'
  19. Rotate(C, If[0 < j ≤ 1, (j * π / 3), π / 3], C) → C'_2
  20. Rotate(B, If[0 < j ≤ 1, (j * π / 3), π / 3], C) → B'_3
  21. Segment(C'_2, B'_3) → i'
  22. Rotate(A, If[1 < j ≤ 2, ((-(j - 1)) * π / 3), ((-π)) / 3], C) → A'_1
  23. Rotate(C, If[1 < j ≤ 2, ((-(j - 1)) * π / 3), ((-π)) / 3], C) → C'_3
  24. Segment(A'_1, C'_3) → h'_1
  25. Rotate(C, If[1 < j ≤ 2, ((-(j - 1)) * π / 3), ((-π)) / 3], B) → C'_4
  26. Rotate(B, If[1 < j ≤ 2, ((-(j - 1)) * π / 3), ((-π)) / 3], B) → B'_4
  27. Segment(C'_4, B'_4) → i'_1
  28. Polygon(A, C, C'_1) → t2, c'_1, a_2, c_1
  29. Polygon(B, B'_3, C) → t3, c_2, b_1, b'_3
  30. Rotate(A, If[2 < j ≤ 3, ((j - 2) * π / 3), π / 3], A) → A'_2
  31. Rotate(B, If[2 < j ≤ 3, ((-(j - 2)) * π / 3), ((-π)) / 3], A) → B'_5
  32. Segment(A'_2, B'_5) → f'
  33. Rotate(A, If[3 < j ≤ 4, ((j - 3) * π / 3), π / 3], B) → A'_3
  34. Rotate(B, If[3 < j ≤ 4, ((-(j - 3)) * π / 3), ((-π)) / 3], B) → B'_6
  35. Segment(A'_3, B'_6) → f'_1
  36. Dilate(C'_1, If[5 < j ≤ 6, 6 - j, 0], B'_5) → F
  37. Vector(C'_1, F) → u
  38. Translate(A'_1, u) → G
  39. Translate(C'_3, u) → H
  40. Dilate(C, If[6 < j ≤ 7, 7 - j, 0], C'_1) → C'_5
  41. Vector(C, C'_5) → v
  42. Translate(C'_2, v) → I
  43. Translate(B'_3, v) → J
  44. Polygon(A, B'_5, B) → t4, b_2, a_3, b'_2
  45. Segment(A, B'_3) → l
  46. Segment(B, C'_1) → m
  47. Min(f, i) → n
  48. Min({f, i, h}) → o
  49. Circle(B, (0.2 * o)) → p
  50. Intersect(p, f, 1) → K
  51. Intersect(p, f'_1) → L,
  52. Intersect(p, m, 1) → M
  53. Circle(A, (0.2 * o)) → q
  54. Line(A, B'_3) → r
  55. Intersect(q, r, 1) → N
  56. Ray(A, N) → s
  57. Intersect(q, f') → O,
  58. Angle(O, A, B'_3) → β
  59. Angle(C'_1, B, L) → γ
  60. Dilate(B'_3, If[4 < j ≤ 5, 5 - j, 0], A) → A'_4
  61. Dilate(C'_1, If[5 < j ≤ 6, 6 - j, 0], B) → B'_7
  62. Rotate(L, If[6 < j ≤ 7, ((-(j - 6)) * γ), (-γ)], B) → L'
  63. CircleSector(B, L', L) → d_1
  64. Rotate(O, If[7 < j ≤ 8, ((j - 7) * β), β], A) → O'
  65. CircleSector(A, O, O') → e_1
  66. Midpoint(A, B) → P
  67. Line(P, s) → f_1
  68. Line(B, s) → g_1
  69. Point(g_1) → Q
  70. Angle(C'_1, B, Q) → δ
  71. Dilate(B, If[8 < j ≤ 9, 9 - j, 0], P) → B'_8
  72. Vector(B, B'_8) → w
  73. Translate(d_1, w) → h_1
  74. Rotate(h_1, If[9 < j ≤ 10, ((j - 9) * π / 3), π / 3], P) → k_1
  75. Dilate(P, If[10 < j ≤ 11, 11 - j, 0], A) → P'
  76. Vector(P, P') → u_1
  77. Translate(k_1, u_1) → p_1
  78. Segment(A, B) → i_1
  79. Segment(B'_3, A'_4) → j_1
  80. Segment(C'_1, B'_7) → t