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Inscribed Angle Theorem Proof Exercise (Case 2)

Inscribed Angle Theorem Proof Exercise (Case 2)
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2017-07-25 23:01
CircleAngleTrigonometryGeometric Construction
In collections Circles (Theorems) Inscribed Angle Theorem: Discovery and Proof
This example demonstrates the Inscribed Angle Theorem for the second case where the center of the circle lies inside the inscribed angle. It involves core mathematical concepts such as circle, inscribed angle, central angle, and geometric proof. Construction highlights include using GeoGebra commands like Circle, Point, Line, Intersect, CircleArc, Segment, CircleSector, Dilate, Angle, Rotate, Polygon, Vector, and Translate to build the geometric figures. Suitable for high school mathematics education to help students understand and prove different cases of the Inscribed Angle Theorem.

GeoGebra Commands & Parameters

  1. Circle(O, B_1) → c
  2. Point(c) → A
  3. Line(A, O) → f
  4. Intersect(c, f, 2) → D_1
  5. CircleArc(O, D_1, A) → d
  6. CircleArc(O, A, D_1) → e
  7. Point(e) → B
  8. Point(d) → C
  9. Segment(A, B) → g
  10. Segment(A, C) → h
  11. Circle(A, (0.25 * Distance[O, B_1])) → k
  12. Intersect(k, g, 1) → G
  13. Intersect(k, h, 1) → H
  14. CircleSector(A, G, H) → p
  15. Dilate(A, If[0 < a ≤ 1, 1 - a, 0], O) → C'
  16. Segment(A, C') → i
  17. Dilate(O, If[1 < a ≤ 2, 2 - a, 0], D_1) → A'
  18. Segment(O, A') → j
  19. Dilate(O, If[3 < a ≤ 4, 4 - a, 0], B) → A'_1
  20. Segment(O, A'_1) → l
  21. Angle(A, O, B) → α
  22. Angle(C, O, A) → β
  23. Rotate(O, If[4 < a ≤ 5, ((-(a - 4)) * α), (-α)], O) → A'_2
  24. Rotate(A'_1, If[4 < a ≤ 5, ((-(a - 4)) * α), (-α)], O) → I
  25. Segment(A'_2, I) → l'
  26. Rotate(O, If[5 < a ≤ 6, ((-(a - 5)) * β), (-β)], O) → A'_3
  27. Rotate(A'_2, If[5 < a ≤ 6, ((-(a - 5)) * β), (-β)], O) → J
  28. Rotate(I, If[5 < a ≤ 6, ((-(a - 5)) * β), (-β)], O) → I'
  29. Segment(J, I') → l''
  30. Circle(B, (0.25 * Min[g, l])) → q
  31. Intersect(q, g) → K,
  32. Intersect(q, l) → L,
  33. CircleSector(B, L, K) → r
  34. Angle(O, B, A) → γ
  35. Rotate(K, If[6 < a ≤ 7, ((-(a - 6)) * γ), (-γ)], B) → K'
  36. CircleSector(B, K', K) → s
  37. Polygon(B, O, A) → poly1, c_1, e_1, a_1
  38. Polygon(O, C, A) → poly2, c_2, a_2, f_1
  39. Dilate(A, If[0 < m ≤ 1, 1 - m, 0], O) → C'_1
  40. Vector(A, C'_1) → u
  41. Translate(p, u) → p'
  42. Line(O, g) → n
  43. Line(O, h) → t
  44. Intersect(d, t) → M, E
  45. Angle(M, O, C) → δ
  46. Rotate(p', If[1 < m ≤ 2, ((m - 1) * δ), δ], O) → p''
  47. Intersect(e, n) → N, D
  48. Angle(B, O, N) → ε
  49. Dilate(A, If[2 < m ≤ 3, 3 - m, 0], O) → C'_2
  50. Vector(A, C'_2) → v
  51. Translate(p, v) → p'_1
  52. Circle(O, Radius[k]) → g_1
  53. Rotate(n, δ, O) → n'
  54. Rotate(p'_1, If[2 < m ≤ 3, ((-(m - 2)) * ε), (-ε)], C'_2) → d_1
  55. CircleArc(O, B, C) → h_1
  56. CircleArc(O, C, B) → k_1