积极 AI Logo 积极 AI

仅对等边三角形成立的动态几何演示

仅对等边三角形成立的动态几何演示
几何画板及绘图内容加载中，请稍等……
2016-08-29 03:47
测试改名0多边形变换几何作图
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
这个范例通过动态几何图形演示了仅在等边三角形情况下成立的特定性质。核心数学概念包括等边三角形、几何变换、外接圆与内接圆、旋转、缩放与对称性。作图要点涉及多边形、外接圆、交点、旋转、平移、向量、中心点、圆弧及 Dilate 等 GeoGebra 命令，通过点与线的动态关联展示等边三角形的不变性。适用学段：高中几何、数学建模启蒙及 GeoGebra 交互学习，帮助学生直观理解等边三角形在动态变换下的特殊性质。

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. Rotate(A, If[3.5 < d ≤ 4.5, ((d - 3.5) * π / 3), 1.0471975511965976], A) → A'_1
  15. Rotate(A', If[3.5 < d ≤ 4.5, ((d - 3.5) * π / 3), 1.0471975511965976], A) → A''
  16. Segment(A'_1, A'') → e'
  17. Rotate(A, If[4.5 < d ≤ 5.5, ((-(d - 4.5)) * π / 3), -1.0471975511965976], B) → A'_2
  18. Rotate(A', If[4.5 < d ≤ 5.5, ((-(d - 4.5)) * π / 3), -1.0471975511965976], B) → A''_1
  19. Segment(A'_2, A''_1) → e'_1
  20. Dilate(A, If[6 < d ≤ 7, 7 - d, 0], B) → A'_3
  21. Vector(A, A'_3) → u
  22. Circle(B, (0.15 * h)) → r
  23. Intersect(f, e'_1) → G
  24. Intersect(f, e'_1) → H
  25. Intersect(f, e'_1) → I
  26. Intersect(f, e'_1) → J
  27. Intersect(r, j, 1) → F
  28. Intersect(r, h, 1) → K
  29. Dilate(B, If[8 < d ≤ 9, 9 - d, 0], C) → B'_1
  30. Vector(B, B'_1) → v
  31. Centroid(poly1) → L
  32. Circle(L, B) → p
  33. CircleArc(L, B, A) → q
  34. CircleArc(L, A, B) → t
  35. Point(t) → M
  36. Rotate(B, If[5.5 < d ≤ 6.5, (((d - 5.5) * 4) * π / 3), (4 * π / 3)], L) → B'_2
  37. CircleArc(L, B, B'_2) → c_1
  38. Rotate(A, If[6.5 < d ≤ 7.5, (((d - 6.5) * 2) * π / 3), (2 * π / 3)], L) → A'_4
  39. CircleArc(L, A, A'_4) → d_1
  40. Segment(A, M) → l
  41. Segment(B, M) → m
  42. Segment(C, M) → n
  43. Circle(M, l) → s
  44. Intersect(s, n) → N,
  45. Dilate(A, If[7.5 < d ≤ 8.5, 8.5 - d, 0], M) → A'_5
  46. Segment(A, A'_5) → f_1
  47. Dilate(B, If[8.5 < d ≤ 9.5, 9.5 - d, 0], M) → B'_3
  48. Segment(B, B'_3) → g_1
  49. Dilate(C, If[9.5 < d ≤ 10.5, 10.5 - d, 0], N) → C'_1
  50. Segment(C, C'_1) → h_1
  51. Dilate(N, If[10.5 < d ≤ 11.5, 11.5 - d, 0], M) → N'
  52. Segment(N, N') → i_1
  53. Circle(L, B) → e_1
  54. Rotate(A, If[11.5 < d ≤ 12.5, ((-(d - 11.5)) * π / 3), ((-π)) / 3], M) → A'_6
  55. Rotate(A'_5, If[11.5 < d ≤ 12.5, ((-(d - 11.5)) * π / 3), ((-π)) / 3], M) → O
  56. Segment(A'_6, O) → j_1
  57. Segment(A, M) → k_1
  58. Dilate(M, If[12.5 < d ≤ 13.5, 13.5 - d, 0], N) → M'
  59. Vector(M, M') → w
  60. Translate(B, w) → B'_4
  61. Translate(B'_3, w) → P
  62. Segment(B'_4, P) → l_1
  63. Segment(B, M) → m_1
  64. Rotate(B'_4, If[13.5 < d ≤ 14.5, ((d - 13.5) * π / 3), π / 3], N) → Q
  65. Rotate(P, If[13.5 < d ≤ 14.5, ((d - 13.5) * π / 3), π / 3], N) → P'
  66. Segment(Q, P') → n_1