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菱形对角线 by Tim Brzezinski

菱形对角线 by Tim Brzezinski
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2016-05-27 11:11
角解析几何
本范例通过解析几何证明菱形对角线互相垂直。它涉及菱形、三角形、比例等核心数学概念。通过使用Circle, Line, Intersect, CircleArc, Point, Segment, Min, Dilate, Angle, Rotate, Polygon, Vector, CircleSector等GeoGebra命令，可以构造出菱形并验证其对角线性质。适合初中及以上学生学习几何证明。

GeoGebra 命令及参数

  1. Circle(B, A) → f
  2. Circle(A, B) → c
  3. Line(A, B) → b
  4. Intersect(c, b, 2) → C
  5. CircleArc(A, B, C) → d
  6. Point(d) → D
  7. Segment(D, B) → s
  8. Circle(D, A) → e
  9. Intersect(e, f, 2) → E
  10. Segment(A, E) → n
  11. Intersect(n, s) → L
  12. Segment(A, B) → a
  13. Segment(A, D) → g
  14. Segment(D, E) → h
  15. Segment(E, B) → i
  16. Min(g, h) → j
  17. Dilate(B, If[0 < l ≤ 1, 1 - l, 0], A) → B'
  18. Segment(B, B') → m
  19. Angle(E, B, A) → α
  20. Angle(D, E, B) → β
  21. Rotate(B, If[1 < l ≤ 2, ((-(l - 1)) * α), α], B) → B'_1
  22. Rotate(B', If[1 < l ≤ 2, ((-(l - 1)) * α), (-α)], B) → B''
  23. Segment(B'_1, B'') → m'
  24. Rotate(B'_1, If[2 < l ≤ 3, ((-(l - 2)) * β), (-β)], E) → J
  25. Rotate(B'', If[2 < l ≤ 3, ((-(l - 2)) * β), (-β)], E) → B'''
  26. Segment(J, B''') → m''
  27. Rotate(J, If[3 < l ≤ 4, ((-(l - 3)) * α), (-α)], D) → J'
  28. Rotate(B''', If[3 < l ≤ 4, ((-(l - 3)) * α), (-α)], D) → K
  29. Segment(J', K) → m'''
  30. Circle(B, (0.2 * j)) → k
  31. Circle(E, (0.2 * j)) → p
  32. Intersect(p, m'', 1) → F
  33. Intersect(p, m', 1) → G
  34. Intersect(k, m', 1) → H
  35. Intersect(k, m, 1) → I
  36. Polygon(A, B, E, D) → poly1, a_1, b_1, e_1, d_1
  37. Dilate(B, If[5 < l ≤ 6, 6 - l, 0], D) → B'_2
  38. Vector(B, B'_2) → u
  39. Dilate(E, If[7 < l ≤ 8, 8 - l, 0], A) → E'
  40. Vector(E, E') → v
  41. Dilate(D, If[4 < l ≤ 5, 5 - l, 0], B) → D'
  42. Vector(D, D') → w
  43. Min(n / 2, s / 2) → o
  44. Circle(L, (0.2 * o)) → t
  45. Intersect(t, s, 2) → M
  46. Intersect(t, n, 2) → N
  47. CircleSector(L, M, N) → c_1
  48. Segment(D, D') → f_1
  49. Dilate(L, If[10 < l ≤ 11, 11 - l, 0], B) → L'
  50. Circle(D, (0.2 * j)) → h_1
  51. Intersect(k, s) → P,
  52. Intersect(s, f_1) → Q
  53. Intersect(h_1, s, 1) → O
  54. Circle(A, (0.2 * j)) → k_1
  55. Intersect(k_1, m, 1) → R
  56. Intersect(k_1, n, 1) → S
  57. Intersect(k_1, m''', 1) → T
  58. Intersect(h_1, m'', 1) → U
  59. Dilate(A, If[5 < l ≤ 6, 6 - l, 0], E) → A'
  60. Segment(A, A') → i_1
  61. Dilate(L, If[12 < l ≤ 13, 13 - l, 0], E) → L'_1
  62. Intersect(p, m'', 1) → V
  63. Intersect(p, n, 1) → W
  64. Intersect(p, m', 1) → Z
  65. Rotate(L, If[13 < l ≤ 14, ((-(l - 13)) * π), -3.141592653589793], L) → L'_2
  66. Rotate(L', If[13 < l ≤ 14, ((-(l - 13)) * π), -3.141592653589793], L) → L''
  67. Rotate(L, If[14 < l ≤ 15, ((-(l - 14)) * π), 3.141592653589793], L) → L'_3
  68. Rotate(L'_1, If[14 < l ≤ 15, ((-(l - 14)) * π), 3.141592653589793], L) → A_1
  69. Rotate(c_1, If[6.25 < l ≤ 7.25, ((l - 6.25) * π / 2), 1.5708], L) → f_2