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圆中角度关系：弦、割线、切线

圆中角度关系：弦、割线、切线
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2014-05-14 05:45
圆角几何作图
所属专题 Circles
①本范例演示圆中弦、割线、切线形成的角度与弧度关系；②涉及核心数学概念包括圆周角定理、圆心角定理、弦切角定理、弧度制等；③作图要点运用了 GeoGebra 命令：Circle（圆）、Line（线）、OrthogonalLine（垂线）、Intersect（交点）、Rotate（旋转）、CircleArc（圆弧）、Point（点）、Segment（线段）、Angle（角度）、CircleSector（扇形）、Dilate（缩放）；④适合高中数学教学，用于帮助学生直观理解圆中角度与弧度的计算与证明。

GeoGebra 命令及参数

  1. Circle(A, 5) → c
  2. Line(A, a) → b
  3. OrthogonalLine(A, a) → d
  4. Intersect(c, d, 2) → B
  5. Rotate(B, 45*°, A) → B'
  6. Rotate(B, -45*°, A) → B'_1
  7. OrthogonalLine(B', a) → e
  8. OrthogonalLine(B'_1, a) → f
  9. Intersect(c, e, 1) → C
  10. Intersect(c, f, 1) → D
  11. CircleArc(A, B'_1, B') → g
  12. CircleArc(A, C, D) → h
  13. Point(g) → E
  14. Point(h) → F
  15. CircleArc(A, D, B'_1) → k
  16. Point(k) → G
  17. Point(k) → H
  18. Line(E, H) → i
  19. Line(G, F) → j
  20. Intersect(i, j) → P
  21. Segment(P, E) → l
  22. Segment(P, G) → m
  23. Segment(P, H) → n
  24. Segment(P, F) → p
  25. Angle(H, A, G) → Arc_2
  26. Angle(E, A, F) → Arc_1
  27. Rotate(E, (-Arc_2), A) → E'
  28. Angle(H, A, G) → γ
  29. Angle(E, P, F) → Angle
  30. CircleArc(A, E, F) → q
  31. CircleArc(A, E', E) → r
  32. CircleArc(A, E, E') → s
  33. CircleSector(A, E, F) → t
  34. Dilate(A, 0.05, E) → A'
  35. Rotate(A', (-Arc_2), A) → A''
  36. CircleArc(A, E, E') → c_1
  37. CircleArc(A, E, A'') → d_1
  38. CircleArc(A, A', A'') → e_1
  39. Dilate(A, 0.05, F) → A'_1
  40. Rotate(A'_1, (-Angle), A) → L
  41. Rotate(L, (-Angle), A) → L'
  42. CircleArc(A, H, G) → f_1
  43. CircleArc(A, G, H) → g_1
  44. Circle(P, Segment[A, A'_1]) → h_1
  45. Intersect(h_1, p, 1) → M
  46. Intersect(h_1, l, 1) → N
  47. CircleArc(P, N, M) → k_1
  48. CircleArc(A, L, F) → p_1
  49. CircleArc(A, L', L) → q_1
  50. Segment(L, A) → a_1
  51. Segment(F, A) → b_1
  52. Segment(L', A) → i_1
  53. Segment(E, A) → j_1
  54. Angle(L', A, L) → ε
  55. Angle(L, A, F) → ζ
  56. Segment(A, F) → l_1
  57. Circle(A_1, 5) → c_2
  58. Line(A_1, a_2) → b_3
  59. OrthogonalLine(A_1, a_2) → d_2
  60. Intersect(c_2, d_2, 2) → B_1
  61. Rotate(B_1, 45*°, A_1) → B'_2
  62. Rotate(B_1, -45*°, A_1) → B'_3
  63. OrthogonalLine(B'_2, a_2) → e_2
  64. OrthogonalLine(B'_3, a_2) → f_2
  65. Intersect(c_2, e_2, 1) → C_1
  66. Intersect(c_2, f_2, 1) → D_1
  67. CircleArc(A_1, B'_3, B'_2) → g_2
  68. CircleArc(A_1, C_1, D_1) → h_2
  69. Point(g_2) → E_1
  70. Point(h_2) → F_1
  71. CircleArc(A_1, D_1, B'_3) → k_2
  72. Angle(E_1, A_1, F_1) → β_1
  73. CircleArc(A_1, E_1, F_1) → q_2
  74. CircleSector(A_1, E_1, F_1) → t_1
  75. Segment(F_1, A_1) → b_2
  76. Segment(E_1, A_1) → j_2
  77. Segment(A_1, F_1) → l_2