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Inscribed Angle Theorem Proof Exercise (Case 1)

Inscribed Angle Theorem Proof Exercise (Case 1)
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2017-07-25 22:56
CircleAngleGeometric Construction
In collections Circles (Theorems) Inscribed Angle Theorem: Discovery and Proof
This example demonstrates the proof of the Inscribed Angle Theorem when the center of the circle lies on the inscribed angle's chord. It involves core mathematical concepts such as circles, inscribed angles, central angles, and geometric proofs. The construction uses GeoGebra commands like Circle, Point, CircleArc, Line, Intersect, Segment, Angle, Rotate, CircleSector, Dilate, Vector, Polygon, and Translate. It is suitable for high school mathematics education, helping students understand and master the proof method of the Inscribed Angle Theorem.

GeoGebra Commands & Parameters

  1. Circle(O, B_1) → c
  2. Point(c) → B
  3. Point(c) → C_1
  4. CircleArc(O, B, C_1) → d
  5. Point(d) → A
  6. Line(A, O) → f_1
  7. Intersect(c, f_1) → Q, C
  8. Point(d) → F
  9. Segment(O, C_1) → a
  10. Segment(A, B) → g
  11. Segment(F, C_1) → h
  12. Segment(F, B) → i
  13. Angle(C_1, O, B) → α
  14. Rotate(C_1, α / 2, O) → C'
  15. Angle(C_1, O, C') → β
  16. Segment(O, C') → j
  17. Segment(A, C) → f
  18. Circle(A, (0.25 * Distance[O, B_1])) → k
  19. Intersect(k, f, 1) → G
  20. Intersect(k, g, 1) → H
  21. Circle(F, (0.25 * Distance[O, B_1])) → p
  22. Intersect(p, h, 1) → J
  23. Intersect(p, i, 1) → K
  24. CircleSector(A, G, H) → q
  25. CircleSector(F, J, K) → r
  26. Dilate(A, If[0 < l ≤ 1, 1 - l, 0], O) → E'
  27. Vector(A, E') → u
  28. Line(A, a) → m
  29. Intersect(k, m, 2) → L
  30. Angle(L, A, G) → γ
  31. Dilate(A, If[2 < l ≤ 3, 3 - l, 0], O) → E'_1
  32. Vector(A, E'_1) → v
  33. Line(A, j) → n
  34. Intersect(k, n, 2) → M
  35. Angle(M, A, G) → δ
  36. Circle(O, (0.25 * Distance[O, B_1])) → t
  37. Intersect(t, a, 1) → N
  38. Intersect(t, j, 1) → O_1
  39. Intersect(t, c, 1) → I
  40. Intersect(k, f_1) → S, T
  41. CircleArc(O, C, B) → e
  42. Intersect(t, f_1) → U, V
  43. Angle(G, A, H) → ε
  44. Segment(O, C) → s
  45. Segment(A, O) → g_1
  46. Dilate(A, If[-1 < l ≤ 0, (-l), 0], O) → E'_2
  47. Segment(A, E'_2) → i_1
  48. Dilate(O, If[0 < l ≤ 1, 1 - l, 0], C) → A'
  49. Segment(O, A') → j_1
  50. Angle(C, O, B) → η
  51. Rotate(O, If[1 < l ≤ 2, ((l - 1) * η), η], O) → A'_1
  52. Rotate(A', If[1 < l ≤ 2, ((l - 1) * η), η], O) → A''
  53. Segment(A'_1, A'') → k_1
  54. Rotate(A'_1, If[2 < l ≤ 3, ((l - 2) * (π - η)), π - η], O) → P
  55. Rotate(A'', If[2 < l ≤ 3, ((l - 2) * (π - η)), π - η], O) → A'''
  56. Segment(P, A''') → b
  57. Polygon(O, B, A) → poly1, e_1, a_1, d_1
  58. Circle(B, Radius[k]) → c_1
  59. Intersect(c_1, k_1) → W,
  60. Angle(A, B, O) → ζ
  61. Rotate(W, If[3 < l ≤ 4, ((-(l - 3)) * ζ), (-ζ)], B) → W'
  62. CircleSector(B, W', W) → h_1
  63. CircleArc(O, B, C) → p_1
  64. Dilate(A, If[0 < o ≤ 1, 1 - o, 0], O) → E'_3
  65. Vector(A, E'_3) → w
  66. Translate(q, w) → q'
  67. Line(A, k_1) → l_1
  68. Point(l_1) → Z
  69. Angle(B, A, Z) → θ
  70. Rotate(q', If[1 < o ≤ 2, ((o - 1) * θ), θ], O) → q''
  71. Dilate(A, If[2 < o ≤ 3, 3 - o, 0], O) → E'_4
  72. Vector(A, E'_4) → u_1
  73. Translate(q, u_1) → q'_1