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任意三角形两边上构造的两个等边三角形关系

任意三角形两边上构造的两个等边三角形关系
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2017-08-18 19:53
测试改名0
该范例演示了在任意三角形的两边上分别构造两个等边三角形时，所出现的特殊关系。核心数学概念包括等边三角形和全等。构造要点包括使用GeoGebra命令如Segment, Rotate, Polygon等。适合初中及以上学生学习几何变换和全等三角形的相关知识。

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(B, C) → h
  7. Segment(A, C) → i
  8. Polygon(A, B, C) → poly1, c, a_1, b
  9. Polygon(C, B, 3) → poly2, j, k, l, D
  10. Polygon(A, C, 3) → poly3, m, n, p, E
  11. Midpoint(C, E) → F
  12. Midpoint(C, D) → G
  13. Midpoint(A, B) → H
  14. Rotate(A, If[0 < d ≤ 1, (d * π / 3), π / 3], A) → A'
  15. Rotate(C, If[0 < d ≤ 1, (d * π / 3), π / 3], A) → C'
  16. Segment(A', C') → i'
  17. Rotate(B, If[0 < d ≤ 1, (d * π / 3), π / 3], C) → B'_1
  18. Rotate(C, If[0 < d ≤ 1, (d * π / 3), π / 3], C) → C'_1
  19. Segment(B'_1, C'_1) → h'
  20. Rotate(B'_1, If[1 < d ≤ 2, ((d - 1) * π / 3), π / 3], D) → I
  21. Rotate(C'_1, If[1 < d ≤ 2, ((d - 1) * π / 3), π / 3], D) → J
  22. Segment(I, J) → h''
  23. Rotate(A', If[1 < d ≤ 2, ((d - 1) * π / 3), π / 3], E) → A''
  24. Rotate(C', If[1 < d ≤ 2, ((d - 1) * π / 3), π / 3], E) → C''
  25. Segment(A'', C'') → i''
  26. Dilate(A, If[2 < d ≤ 3, 3 - d, 0], H) → A'_1
  27. Segment(A, A'_1) → q
  28. Rotate(A, If[3 < d ≤ 4, ((-(d - 3)) * π), (-π)], H) → A'_2
  29. Rotate(A'_1, If[3 < d ≤ 4, ((-(d - 3)) * π), (-π)], H) → K
  30. Segment(A'_2, K) → q'
  31. Dilate(E, If[4 < d ≤ 5, 5 - d, 0], F) → E'
  32. Dilate(D, If[4 < d ≤ 5, 5 - d, 0], G) → D'
  33. Segment(E, E') → r
  34. Segment(D, D') → s
  35. Rotate(E, If[5 < d ≤ 6, ((d - 5) * π), π], F) → E'_1
  36. Rotate(E', If[5 < d ≤ 6, ((d - 5) * π), π], F) → E''
  37. Segment(E'_1, E'') → r'
  38. Rotate(D, If[5 < d ≤ 6, ((-(d - 5)) * π), (-π)], G) → D'_1
  39. Rotate(D', If[5 < d ≤ 6, ((-(d - 5)) * π), (-π)], G) → D''
  40. Segment(D'_1, D'') → s'
  41. Polygon(G, H, F) → poly4, f_1, g_1, h_1
  42. Dilate(G, If[6 < d ≤ 7, 7 - d, 0], H) → G'
  43. Segment(G, G') → t
  44. Rotate(G, If[7 < d ≤ 8, ((d - 7) * π / 3), π / 3], H) → G'_1
  45. Rotate(G', If[7 < d ≤ 8, ((d - 7) * π / 3), π / 3], H) → G''
  46. Segment(G'_1, G'') → t'
  47. Rotate(G'_1, If[8 < d ≤ 9, ((d - 8) * π / 3), π / 3], F) → L
  48. Rotate(G'', If[8 < d ≤ 9, ((d - 8) * π / 3), π / 3], F) → G'''
  49. Segment(L, G''') → t''
  50. Rotate(G'_1, If[8 < d ≤ 9, ((d - 8) * π / 3), π / 3], F) → M
  51. Rotate(G'', If[8 < d ≤ 9, ((d - 8) * π / 3), π / 3], F) → G'''_1
  52. Segment(M, G'''_1) → t''_1