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动态等边三角形与中值问题演示

动态等边三角形与中值问题演示
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2018-07-04 11:00
测试改名0函数几何作图
①本范例动态演示了Antonio Gutierrez在Twitter上提出的一个关于等边三角形、中值、指数函数、面积、余弦及菱形的问题；②涉及核心数学概念包括等边三角形构造、中位线性质、指数函数图像、面积计算、余弦定理、菱形特性；③作图要点提炼自GeoGebra命令：Polygon(多边形), Line(线), OrthogonalLine(垂线), Intersect(交点), Midpoint(中点), Segment(线段), Point(点), Rotate(旋转), Angle(角度), Ray(射线), Dilate(缩放), Min(最小值), Circle(圆), CircleSector(圆扇形), Vector(向量), Translate(平移)；④适合高中数学教学，用于探究几何变换与函数结合问题。

GeoGebra 命令及参数

  1. Polygon(A, B, 3) → poly1, f, g, h, C
  2. Line(C, A) → k
  3. OrthogonalLine(A, g) → i
  4. OrthogonalLine(B, h) → j
  5. Intersect(i, j) → D
  6. Intersect(h, j) → E
  7. Midpoint(C, B) → F
  8. Segment(C, F) → l
  9. Point(l) → G
  10. Rotate(G, 60°, D) → G'
  11. Angle(G, D, G') → α
  12. Ray(D, G') → m
  13. Intersect(h, m) → H
  14. Polygon(H, G, C) → t1, c, h_1, g_1
  15. Polygon(A, B, G, H) → q1, a_1, b, g_2, h_2
  16. Angle(A, F, B) → β
  17. Angle(A, E, B) → γ
  18. Dilate(A, If[0 < d ≤ 1, 1 - d, 0], B) → A'
  19. Segment(A, A') → n
  20. Rotate(A, If[1 < d ≤ 2, ((d - 1) * π / 3), π / 3], A) → A'_1
  21. Rotate(A', If[1 < d ≤ 2, ((d - 1) * π / 3), π / 3], A) → A''
  22. Segment(A'_1, A'') → n'
  23. Rotate(A'_1, If[2 < d ≤ 3, ((d - 2) * π / 3), π / 3], C) → I
  24. Rotate(A'', If[2 < d ≤ 3, ((d - 2) * π / 3), π / 3], C) → A'''
  25. Segment(I, A''') → n''
  26. Dilate(A, If[3 < d ≤ 4, 4 - d, 0], F) → J
  27. Segment(A, J) → p
  28. Dilate(B, If[4 < d ≤ 5, 5 - d, 0], D) → B'
  29. Segment(B, B') → q
  30. Dilate(D, If[5 < d ≤ 6, 6 - d, 0], E) → D'
  31. Segment(D, D') → r
  32. Segment(C, A) → s
  33. Segment(D, G) → t
  34. Segment(D, H) → e
  35. Segment(C, G) → f_1
  36. Segment(C, H) → i_1
  37. Min({e, t, f_1, i_1}) → o
  38. Circle(C, (0.45 * o)) → c_1
  39. Intersect(s, i_1) → K
  40. Intersect(c_1, i_1, 1) → L
  41. Rotate(L, If[9 < d ≤ 10, ((d - 9) * π / 3), π / 3], C) → L'
  42. CircleSector(C, L, L') → d_1
  43. Dilate(D, If[9 < d ≤ 10, 10 - d, 0], G) → D'_1
  44. Dilate(D, If[9 < d ≤ 10, 10 - d, 0], H) → D'_2
  45. Segment(D, D'_1) → j_1
  46. Segment(D, D'_2) → k_1
  47. Line(C, j_1) → l_1
  48. Line(C, k_1) → m_1
  49. Point(l_1) → M
  50. Angle(L, C, M) → δ
  51. Dilate(C, If[11 < d ≤ 12, 12 - d, 0], D) → C'
  52. Vector(C, C') → u
  53. Translate(d_1, u) → e_1
  54. Rotate(e_1, If[11 < d ≤ 12, ((d - 11) * δ), δ], C') → p_1
  55. Dilate(C, If[12 < d ≤ 13, 13 - d, 0], H) → C'_1
  56. Dilate(G, If[12 < d ≤ 13, 13 - d, 0], C) → C'_2
  57. Dilate(H, If[12 < d ≤ 13, 13 - d, 0], G) → H'
  58. Segment(H, H') → n_1
  59. Segment(G, C'_2) → q_1
  60. Segment(C, C'_1) → r_1
  61. Polygon(H, G, C) → t2, c_2, h_3, g_3
  62. Angle(G, H, C) → ε
  63. Line(A, B) → s_1
  64. Line(G, s_1) → t_1
  65. Point(t_1) → N
  66. Angle(H, G, N) → ζ
  67. Dilate(H, If[14 < d ≤ 15, 15 - d, 0], A) → H'_1
  68. Vector(H, H'_1) → v
  69. Translate(H, v) → H'_2
  70. Translate(H, v) → H'_3
  71. Translate(H', v) → H''
  72. Segment(H'_3, H'') → b_1
  73. Rotate(H'_3, If[14 < d ≤ 15, ((d - 14) * ζ), ζ], H'_1) → O
  74. Rotate(H'', If[14 < d ≤ 15, ((d - 14) * ζ), ζ], H'_1) → H'''
  75. Segment(O, H''') → f_2
  76. Dilate(H, If[15 < d ≤ 16, 16 - d, 0], H''') → H'_4
  77. Vector(H, H'_4) → w
  78. Translate(C, w) → C'_3
  79. Translate(C'_1, w) → P
  80. Segment(C'_3, P) → i_2
  81. Rotate(C'_3, If[15 < d ≤ 16, ((-(d - 15)) * π / 3), ((-π)) / 3], P) → Q
  82. Rotate(P, If[15 < d ≤ 16, ((-(d - 15)) * π / 3), ((-π)) / 3], P) → P'
  83. Segment(Q, P') → j_2
  84. Dilate(C, If[16 < d ≤ 17, 17 - d, 0], Q) → G'_1
  85. Vector(C, G'_1) → u_1
  86. Translate(G, u_1) → G'_2
  87. Translate(C'_2, u_1) → R
  88. Segment(G'_2, R) → k_2
  89. Rotate(G'_2, If[16 < d ≤ 17, ((d - 16) * π / 3), π / 3], R) → S
  90. Rotate(R, If[16 < d ≤ 17, ((d - 16) * π / 3), π / 3], R) → R'
  91. Segment(S, R') → l_2