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等边三角形分割成矩形与杜登尼 puzzle

等边三角形分割成矩形与杜登尼 puzzle
几何画板及绘图内容加载中，请稍等……
2015-07-24 19:03
测试改名0多边形向量解析几何几何作图
①本范例演示如何将等边三角形分割成矩形，并探究分割点的位置关系；②涉及的核心数学概念包括等边三角形、矩形、坐标几何、向量、旋转、几何变换等；③作图要点提炼自构造摘要，使用了 UnitVector, Segment, Intersect, Point, If, Circle, Rotate, Polygon, Angle, UnitOrthogonalVector, Distance, OrthogonalVector, Translate, Midpoint 等GeoGebra命令和对象；④适合高中及以上数学学习者，用于理解几何分割与变换的原理，提升空间想象能力。

GeoGebra 命令及参数

  1. UnitVector(Vector[A, B]) → u
  2. Segment(A, B) → c
  3. Segment(C, B) → a
  4. Intersect(OrthogonalLine[Midpoint[a], c], c) → G'
  5. Point(Segment[A, G']) → E
  6. If(x(E) < x(A + c / 2 * u), E + c / 2 * u, B) → M
  7. If(x(E) < x(A) + c / 2, {Segment[E, M - 0.03 * c * u], M, Segment[M + 0.03 * c * u, B]}, {Segment[E, B]}) → list1
  8. Segment(A, C) → b
  9. Intersect(OrthogonalLine[Midpoint[b], c], c) → G
  10. Circle(Midpoint[a], sqrt(Distance[A, a] * a / 2)) → k_1
  11. Intersect(k_1, c) → S, T
  12. If(γ < 180°, γ, 180*°) → θ
  13. UnitVector(Vector[B, C]) → v
  14. Point(a) → D
  15. Rotate(B, θ, D) → P
  16. If(γ < 180°, 0*°, γ - 180°) → ϕ
  17. Point(list1) → F
  18. Rotate(F, Angle[B, D, P], D) → F'
  19. Rotate(E, Angle[B, D, P], D) → E'
  20. UnitVector(Vector[A, C]) → w
  21. Point(b) → I
  22. Segment(D, E) → d
  23. Rotate(D, Angle[B, D, P], D) → D'
  24. Rotate(B, Angle[B, D, P], D) → B'
  25. Polygon(D', E', B') → poly1, b', d', e'
  26. Intersect(OrthogonalLine[I, d], d) → J
  27. Rotate(A, -θ, I) → K
  28. Rotate(I, Angle[A, I, K], I) → I'
  29. Rotate(J, Angle[A, I, K], I) → J'
  30. Rotate(E, Angle[A, I, K], I) → E'_1
  31. Rotate(A, Angle[A, I, K], I) → A'
  32. Polygon(C, D, J, I) → poly5, c_1, d_2, j, i
  33. Intersect(OrthogonalLine[F, d], d) → L
  34. Rotate(L, Angle[B, D, P], D) → L'
  35. Polygon(P, D, L', F') → poly3, e, f, g, h
  36. Segment(L, F) → k
  37. Segment(I, J) → l
  38. Rotate(E', ϕ, F') → N
  39. Polygon(I', J', E'_1, A') → poly4, i', j', e'_1, a'
  40. Rotate(E', Angle[E', F', N], F') → E''_1
  41. Rotate(L', Angle[E', F', N], F') → L''_1
  42. Rotate(F', Angle[E', F', N], F') → F''_1
  43. Polygon(E''_1, L''_1, F''_1) → poly6, f''_1, e''_1, l''_1
  44. Angle(j, d) → β
  45. Angle(F, L, D) → α
  46. UnitOrthogonalVector(u) → z
  47. Distance(D, A) → DA
  48. UnitVector(Vector[F''_1, L''_1]) → u_1
  49. Segment(L''_1 + 0.15 * u_1, L''_1 + 0.4 * u_1) → a_1
  50. OrthogonalVector(u_1) → v_1
  51. Segment(Midpoint[a_1], Translate[Midpoint[a_1], Vector[(f''_1 + j') * v_1]]) → b_1
  52. Translate(a_1, Vector[(f''_1 + j') * v_1]) → d_1
  53. Midpoint(b_1) → Q
  54. Segment(L''_1 - 0.15 * v_1, L''_1 - 0.4 * v_1) → e_1
  55. Segment(Midpoint[e_1], Translate[Midpoint[e_1], Vector[-(e''_1 + g) * u_1]]) → f_1
  56. Translate(e_1, Vector[-(e''_1 + g) * u_1]) → g_1
  57. Midpoint(f_1) → R
  58. Midpoint(Midpoint[a], B) → H
  59. Midpoint(Midpoint[a], C) → O
  60. UnitOrthogonalVector(Segment[Midpoint[a], Intersect[LineBisector[a], Line[A, a]]]) → n
  61. Intersect(a, OrthogonalLine[A, a]) → U
  62. Segment(I, C) → m
  63. Segment(I, A) → r
  64. Segment(D, C) → p
  65. Segment(D, B) → s
  66. Segment(B, F) → t
  67. Segment(E, F) → h_1
  68. Segment(E, A) → i_1
  69. Segment(E, J) → q
  70. Segment(J, L) → j_1
  71. Segment(L, D) → l_1
  72. Circle(Midpoint[a], sqrt(Distance[A, a] * a / 2)) → p_1
  73. Segment(A, Intersect[a, OrthogonalLine[A, a]]) → m_1
  74. Segment(Midpoint[a], S) → n_1
  75. Angle(A, Intersect[a, OrthogonalLine[A, a]], B) → δ
  76. Segment(I, E) → q_1
  77. Angle(I, E, A) → ε
  78. Segment(F, D) → r_1
  79. Angle(D, F, B) → ζ