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圆内接角定理证明练习（圆心在角外）

圆内接角定理证明练习（圆心在角外）
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2017-07-25 23:07
圆角变换
所属专题 Circles (Theorems) Inscribed Angle Theorem: Discovery and Proof
本范例演示圆内接角定理的证明，特别关注圆心位于内接角外部的情况。通过动态可修改的图形，帮助学生发现和理解几何关系。使用了圆、线、交点、旋转、角度等 GeoGebra 命令。适合初中及以上学生学习几何证明。

GeoGebra 命令及参数

  1. Circle(O, B_1) → c
  2. Point(c) → A
  3. Line(A, O) → f
  4. Intersect(c, f, 2) → D
  5. Semicircle(D, A) → d
  6. Semicircle(A, D) → e
  7. Point(d) → B
  8. CircleArc(O, A, B) → g
  9. CircleArc(O, B, D) → h
  10. Point(h) → C
  11. Segment(A, B) → i
  12. Segment(O, B) → k
  13. Segment(O, C) → l
  14. Segment(B, C) → j
  15. Segment(A, C) → m
  16. Circle(A, (0.25 * Distance[O, A])) → p
  17. Intersect(p, i, 1) → G
  18. Intersect(p, m, 1) → H
  19. CircleSector(A, G, H) → q
  20. Dilate(A, If[0 < a ≤ 1, 1 - a, 0], O) → C'
  21. Segment(A, C') → n
  22. Dilate(O, If[1 < a ≤ 2, 2 - a, 0], D) → A'
  23. Segment(O, A') → r
  24. Rotate(A, If[2 < a ≤ 3, ((-(a - 2)) * π), (-π)], O) → C'_1
  25. Segment(O, C'_1) → s
  26. Angle(C, O, D) → α
  27. Rotate(O, If[3 < a ≤ 4, ((-(a - 3)) * α), (-α)], O) → A'_1
  28. Rotate(C'_1, If[3 < a ≤ 4, ((-(a - 3)) * α), (-α)], O) → I
  29. Segment(A'_1, I) → s'
  30. Circle(C, (0.25 * Distance[O, A])) → t
  31. Intersect(t, l, 1) → J
  32. Angle(O, C, A) → β
  33. Dilate(A, If[4 < a ≤ 5, 5 - a, 0], C) → C'_2
  34. Segment(A, C'_2) → b
  35. Polygon(C, O, A) → poly1, c_1, f_1, a_1
  36. Rotate(J, If[5.5 < a ≤ 6.5, ((a - 5.5) * β), β], C) → J'
  37. CircleSector(C, J, J') → d_1
  38. Angle(A, O, B) → γ
  39. Rotate(A, If[7 < a ≤ 8, ((a - 7) * γ), γ], O) → C'_3
  40. Rotate(C', If[7 < a ≤ 8, ((a - 7) * γ), γ], O) → C''
  41. Segment(C'_3, C'') → n'
  42. Polygon(O, A, B) → poly2, e_1, a_2, c_2
  43. Segment(O, B) → g_1
  44. Circle(B, (0.25 * Distance[O, A])) → h_1
  45. Intersect(h_1, g_1, 1) → K
  46. Angle(O, B, A) → δ
  47. Rotate(K, If[8.5 < a ≤ 9.5, ((a - 8.5) * δ), δ], B) → K'
  48. CircleSector(B, K, K') → k_1
  49. Dilate(A, If[0 < o ≤ 1, 1 - o, 0], O) → C'_4
  50. Vector(A, C'_4) → v
  51. Translate(q, v) → q'
  52. Line(O, b) → i_1
  53. Intersect(h, i_1, 1) → L
  54. Angle(C, O, L) → ε
  55. Rotate(q', If[1 < o ≤ 2, ((-(o - 1)) * ε), (-ε)], O) → q''
  56. Dilate(A, If[2 < o ≤ 3, 3 - o, 0], O) → C'_5
  57. Vector(A, C'_5) → w
  58. Translate(q, w) → q'_1
  59. Line(O, i) → j_1
  60. Intersect(h, j_1, 1) → M
  61. Angle(B, O, M) → ζ
  62. Rotate(q'_1, If[2 < o ≤ 3, ((-(o - 2)) * ζ), (-ζ)], C'_5) → p_1
  63. CircleArc(O, B, C) → q_1
  64. CircleArc(O, C, B) → r_1