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拿破仑运动动态演示

拿破仑运动动态演示
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2016-05-22 03:51
测试改名0多边形几何作图
所属专题 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 Geometry Review Triangles: Points of Concurrency
①本范例动态演示拿破仑定理，通过动态图形展示三个等边三角形的中心构成的三角形始终为等边三角形这一现象；②涉及的核心数学概念包括三角形、等边三角形、中心点、旋转、缩放等几何变换；③作图要点——从下方『构造摘要』提炼用到了 Polygon（多边形）、Segment（线段）、Rotate（旋转）、Midpoint（中点）、Min（最小值）、Dilate（缩放）、Intersect（交点）等 GeoGebra 命令/对象；④适合高中及以上数学学习者，用于理解几何变换与定理证明的直观方法。

GeoGebra 命令及参数

  1. Polygon(A, B, C) → poly1, c, a, b
  2. Polygon(C, B, 3) → poly2, d, e, f, D
  3. Polygon(A, C, 3) → poly3, g, h, i, E
  4. Polygon(B, A, 3) → poly4, j, k, l, F
  5. Segment(A, B) → m
  6. Segment(B, C) → n
  7. Segment(C, A) → p
  8. Rotate(B, If[0 < o ≤ 1, (o * π / 3), 1.0471975511965976], B) → B'
  9. Rotate(C, If[0 < o ≤ 1, ((-o) * π / 3), -1.0472], B) → C'
  10. Segment(B', C') → n'
  11. Rotate(B, If[1 < o ≤ 2, ((o - 1) * π / 3), 1.0472], C) → B'_1
  12. Rotate(C, If[0 < o ≤ 1, (o * π / 3), 1.0471975511965976], C) → C'_1
  13. Segment(B'_1, C'_1) → n'_1
  14. Rotate(C, If[2 < o ≤ 3, ((-(o - 2)) * π / 3), -1.0471975511965976], C) → C'_2
  15. Rotate(A, If[2 < o ≤ 3, ((-(o - 2)) * π / 3), -1.0471975511965976], C) → A'
  16. Segment(C'_2, A') → p'
  17. Rotate(C, If[3 < o ≤ 4, ((o - 3) * π / 3), 1.0471975511965976], A) → C'_3
  18. Rotate(A, If[3 < o ≤ 4, ((o - 3) * π / 3), 1.0471975511965976], A) → A'_1
  19. Segment(C'_3, A'_1) → p'_1
  20. Rotate(A, If[4 < o ≤ 5, ((-(o - 4)) * π / 3), -1.0471975511965976], A) → A'_2
  21. Rotate(B, If[4 < o ≤ 5, ((-(o - 4)) * π / 3), -1.0471975511965976], A) → B'_2
  22. Segment(A'_2, B'_2) → m'
  23. Rotate(A, If[5 < o ≤ 6, ((o - 5) * π / 3), 1.0471975511965976], B) → A'_3
  24. Rotate(B, If[5 < o ≤ 6, ((o - 5) * π / 3), 1.0471975511965976], B) → B'_3
  25. Segment(A'_3, B'_3) → m'_1
  26. Midpoint(p) → G
  27. Midpoint(n) → H
  28. Midpoint(m) → I
  29. Segment(D, H) → q
  30. Segment(E, G) → r
  31. Segment(F, I) → s
  32. Min(list1) → t
  33. Segment(H, B) → a_1
  34. Segment(C, G) → b_1
  35. Segment(A, I) → c_1
  36. Dilate(D, If[6 < o ≤ 7, 7 - o, 0], H) → D'
  37. Segment(D, D') → d_1
  38. Dilate(E, If[6 < o ≤ 7, 7 - o, 0], G) → E'
  39. Segment(E, E') → e_1
  40. Dilate(F, If[6 < o ≤ 7, 7 - o, 0], I) → F'
  41. Segment(F, F') → f_1
  42. Rotate(H, If[7.5 < o ≤ 8.5, ((o - 7.5) * π), 3.141592653589793], H) → H'
  43. Rotate(B, If[7.5 < o ≤ 8.5, ((o - 7.5) * π), 3.141592653589793], H) → B'_4
  44. Segment(H', B'_4) → g_1
  45. Rotate(C, If[7.5 < o ≤ 8.5, ((o - 7.5) * π), 3.141592653589793], G) → C'_4
  46. Rotate(G, If[7.5 < o ≤ 8.5, ((o - 7.5) * π), 3.141592653589793], G) → G'
  47. Segment(C'_4, G') → h_1
  48. Rotate(A, If[7.5 < o ≤ 8.5, ((o - 7.5) * π), 3.141592653589793], I) → A'_4
  49. Rotate(I, If[7.5 < o ≤ 8.5, ((o - 7.5) * π), 3.141592653589793], I) → I'
  50. Segment(A'_4, I') → i_1
  51. Midpoint(p') → J
  52. Midpoint(m') → K
  53. Midpoint(n') → L
  54. Segment(J, C) → j_1
  55. Segment(B, L) → k_1
  56. Segment(A, K) → l_1
  57. Dilate(A, If[9 < o ≤ 10, 10 - o, 0], J) → A'_5
  58. Segment(A, A'_5) → m_1
  59. Dilate(C, If[9 < o ≤ 10, 10 - o, 0], L) → C'_5
  60. Segment(C, C'_5) → n_1
  61. Dilate(B, If[9 < o ≤ 10, 10 - o, 0], K) → B'_5
  62. Segment(B, B'_5) → p_1
  63. Rotate(B, If[10.5 < o ≤ 11.5, ((o - 10.5) * π), 3.141592653589793], L) → B'_6
  64. Rotate(L, If[10.5 < o ≤ 11.5, ((o - 11.5) * π), 3.141592653589793], L) → L'
  65. Segment(B'_6, L') → q_1
  66. Rotate(J, If[10.5 < o ≤ 11.5, ((o - 10.5) * π), 3.141592653589793], J) → J'
  67. Rotate(C, If[10.5 < o ≤ 11.5, ((o - 10.5) * π), 3.141592653589793], J) → C'_6
  68. Segment(J', C'_6) → r_1
  69. Rotate(A, If[10.5 < o ≤ 11.5, ((o - 10.5) * π), 3.141592653589793], K) → A'_6
  70. Rotate(K, If[10.5 < o ≤ 11.5, ((o - 10.5) * π), 3.141592653589793], K) → K'
  71. Segment(A'_6, K') → s_1
  72. Intersect(e_1, m_1) → M
  73. Intersect(f_1, p_1) → N
  74. Intersect(d_1, n_1) → O
  75. Dilate(N, If[12 < o ≤ 13, 13 - o, 0], O) → N'
  76. Segment(N, N') → t_1
  77. Rotate(N, If[13 < o ≤ 14, ((-(o - 13)) * π / 3), -1.0472], O) → N'_1
  78. Rotate(N', If[13 < o ≤ 14, ((o - 13) * π / 3), 1.0471975511965976], O) → N''
  79. Segment(N'_1, N'') → a_2
  80. Rotate(N, If[14 < o ≤ 15, ((o - 14) * π / 3), 1.0471975511965976], N) → N'_2
  81. Rotate(N', If[14 < o ≤ 15, ((o - 14) * π / 3), 1.0471975511965976], N) → N''_1
  82. Segment(N'_2, N''_1) → b_2
  83. Polygon(N, O, M) → poly5, m_2, n_2, o_1