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特殊三角形（二）：30-60-90 三角形与等边三角形的关系

特殊三角形（二）：30-60-90 三角形与等边三角形的关系
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2018-08-14 01:22
测试改名0多边形三角函数变换几何作图
本范例通过无文字、无角度标注的交互式几何作图，引导学生自主发现 30-60-90 三角形恰好是等边三角形的一半。核心数学概念包括特殊直角三角形的边长比例、对称性以及等边三角形的性质。作图要点：使用 GeoGebra 的 Polygon、Dilate、Segment、Circle、Intersect、CircleSector、Rotate、Vector、Translate、CircleArc、Midpoint 等命令，构造了等边三角形及其内切/外接圆、旋转与平移后的辅助图形，并通过多个点（如 A、B、C、D、E、F、G、H、I、J、K）和线段（a、b、c、d、e、f、g、h、i、j、k）以及圆弧（CircleSector、CircleArc）来展示几何关系。适合中学几何教学，用于直观理解 30-60-90 三角形的性质及其与等边三角形的联系，也可用于探究对称和旋转变换。

GeoGebra 命令及参数

  1. Polygon(A, B, 3) → poly1, a, b, c, C
  2. Dilate(A, If[0 < d ≤ 1, 1 - d, 0], B) → A'
  3. Segment(A, A') → e
  4. Dilate(B, If[1 < d ≤ 2, 2 - d, 0], C) → B'
  5. Segment(B, B') → f
  6. Dilate(C, If[2 < d ≤ 3, 3 - d, 0], A) → C'
  7. Segment(C, C') → g
  8. Segment(A, B) → h
  9. Segment(A, C) → i
  10. Segment(B, C) → j
  11. Circle(A, (0.15 * h)) → k
  12. Intersect(k, i, 1) → D
  13. Intersect(k, h, 1) → E
  14. CircleSector(A, E, D) → p
  15. Rotate(A, If[3.5 < d ≤ 4.5, ((d - 3.5) * π / 3), 1.0471975511965976], A) → A'_1
  16. Rotate(A', If[3.5 < d ≤ 4.5, ((d - 3.5) * π / 3), 1.0471975511965976], A) → A''
  17. Segment(A'_1, A'') → e'
  18. Rotate(A, If[4.5 < d ≤ 5.5, ((-(d - 4.5)) * π / 3), -1.0471975511965976], B) → A'_2
  19. Rotate(A', If[4.5 < d ≤ 5.5, ((-(d - 4.5)) * π / 3), -1.0471975511965976], B) → A''_1
  20. Segment(A'_2, A''_1) → e'_1
  21. Dilate(A, If[6 < d ≤ 7, 7 - d, 0], B) → A'_3
  22. Vector(A, A'_3) → u
  23. Translate(p, u) → p'
  24. CircleArc(A, E, D) → q
  25. Rotate(p', If[7 < d ≤ 8, (((d - 7) * 2) * π / 3), 2.0943951023931953], B) → p''
  26. Circle(B, (0.15 * h)) → r
  27. Intersect(f, e'_1) → G
  28. Intersect(f, e'_1) → H
  29. Intersect(f, e'_1) → I
  30. Intersect(f, e'_1) → J
  31. Intersect(r, j, 1) → F
  32. Intersect(r, h, 1) → K
  33. CircleArc(B, F, K) → s
  34. Dilate(B, If[8 < d ≤ 9, 9 - d, 0], C) → B'_1
  35. Vector(B, B'_1) → v
  36. Translate(p'', v) → p'''
  37. Rotate(p''', If[9 < d ≤ 10, (((d - 9) * 2) * π / 3), 2.0943951023931953], C) → t
  38. Circle(C, (0.15 * h)) → c_1
  39. Intersect(g, e') → L
  40. Intersect(c_1, i) → M,
  41. Intersect(c_1, j) → N,
  42. Midpoint(h) → O
  43. Dilate(C, If[10 < d ≤ 11, 11 - d, 0], O) → C'_1
  44. Segment(C, C'_1) → l
  45. Circle(O, (0.15 * h)) → d_1
  46. Segment(C, O) → m
  47. Intersect(d_1, m, 1) → P
  48. Intersect(d_1, h) → Q, R
  49. Polygon(A, O, C) → poly2, c_2, a_1, o
  50. Intersect(c_1, m) → S,
  51. CircleSector(C, S, N) → e_1
  52. CircleSector(O, R, P) → f_1
  53. Segment(O, B) → n
  54. Rotate(f_1, If[11.25 < d ≤ 12.25, ((d - 11.25) * π / 2), 1.5708], O) → g_1
  55. Rotate(e_1, If[12.25 < d ≤ 13.25, ((-(d - 12.25)) * π / 6), -0.5236], C) → h_1
  56. CircleArc(C, S, N) → k_1
  57. CircleArc(O, R, P) → p_1
  58. Rotate(O, If[13.25 < d ≤ 14.25, ((-(d - 13.25)) * π), -3.141592653589793], O) → O'
  59. Rotate(B, If[13.25 < d ≤ 14.25, ((d - 13.25) * π), 3.14159], O) → B'_2
  60. Segment(O', B'_2) → n'
  61. Segment(O, B) → b_1
  62. Dilate(C, If[14.25 < d ≤ 15.25, 15.25 - d, 0], A) → C'_2
  63. Vector(C, C'_2) → w
  64. Translate(h_1, w) → q_1
  65. Rotate(q_1, If[15.25 < d ≤ 16.25, (((-(d - 15.25)) * 4) * π / 3), -4.1887902047863905], A) → r_1
  66. Dilate(C, If[16.25 < d ≤ 17.25, 17.25 - d, 0], A) → C'_3
  67. Vector(C, C'_3) → u_1
  68. Translate(h_1, u_1) → s_1
  69. Rotate(s_1, If[17.25 < d ≤ 18.25, (((-(d - 17.25)) * 7) * π / 6), -3.665191429188092], A) → t_1
  70. Midpoint(g) → T
  71. Rotate(O', If[18.25 < d ≤ 19.25, ((d - 18.25) * π / 3), 1.0471975511965976], A) → O''
  72. Rotate(B'_2, If[18.25 < d ≤ 19.25, ((d - 18.25) * π / 3), 1.0471975511965976], A) → U
  73. Segment(O'', U) → n''
  74. Dilate(A, If[19.25 < d ≤ 20.25, 20.25 - d, 0], T) → A'_4
  75. Vector(A, A'_4) → v_1
  76. Translate(O', v_1) → O''_1
  77. Translate(B'_2, v_1) → V
  78. Segment(O''_1, V) → n''_1
  79. Rotate(O''_1, If[20.25 < d ≤ 21.25, ((d - 20.25) * π / 3), 1.0471975511965976], T) → W
  80. Rotate(V, If[20.25 < d ≤ 21.25, ((d - 20.25) * π / 3), 1.0471975511965976], T) → V'
  81. Segment(W, V') → i_1