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Inscribed Angle Theorem Proof Exercise (Case 3: Center Outside Inscribed Angle)

Inscribed Angle Theorem Proof Exercise (Case 3: Center Outside Inscribed Angle)
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2017-07-25 23:07
CircleAngleTransformation
In collections Circles (Theorems) Inscribed Angle Theorem: Discovery and Proof
This example demonstrates the proof of the Inscribed Angle Theorem, focusing on the case where the center of the circle is outside the inscribed angle. Using dynamic and modifiable geometry, it helps students discover and understand geometric relationships. GeoGebra commands used include Circle, Line, Intersect, Rotate, Angle, etc. Suitable for middle school and above students learning geometry proofs.

GeoGebra Commands & Parameters

  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