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Dynamic Demonstration of a Theorem involving Triangle, Circumcircle, and Angles

Dynamic Demonstration of a Theorem involving Triangle, Circumcircle, and Angles
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2017-03-14 10:15
TriangleCircleAngleGeometric Construction
This example dynamically illustrates a theorem involving a triangle, circumcircle, and three angles. Key math concepts include circumcircle, angles, similarity transformation, and equations. Construction highlights the use of commands like Line, Rotate, Angle, Ray, Segment, Point, Polygon, Circle, Center, Intersect, CircleArc, Min, CircleSector, Dilate, Vector, and Translate in GeoGebra. Suitable for high school mathematics students and teachers for visualizing geometric theorems and exploration.

GeoGebra Commands & Parameters

  1. Line(A, B) → f
  2. Rotate(B, (a * π / 180), A) → B'
  3. Angle(B, A, B') → α
  4. Ray(A, B') → g
  5. Segment(A, B) → h
  6. Point(g) → C
  7. Segment(A, C) → i
  8. Segment(C, B) → j
  9. Point(j) → D
  10. Polygon(A, C, D) → poly1, d, a_1, c
  11. Circle(A, D, C) → e
  12. Center(e) → E
  13. Angle(B, A, D) → β
  14. Rotate(B, β / 2, A) → B'_1
  15. Angle(B, A, B'_1) → γ
  16. Ray(A, B'_1) → k
  17. Intersect(e, k, 2) → F
  18. Intersect(j, k) → G
  19. Ray(C, F) → l
  20. Intersect(h, l) → H
  21. Segment(G, H) → m
  22. Segment(D, H) → n
  23. Segment(D, F) → p
  24. Angle(F, D, H) → δ
  25. Angle(F, C, D) → ε
  26. Angle(G, H, D) → ζ
  27. Rotate(D, If[0 < b ≤ 2, (b * π), (2 * π)], E) → D'
  28. CircleArc(E, D, D') → q
  29. Segment(A, D) → r
  30. Segment(A, F) → s
  31. Segment(A, H) → t
  32. Min({r, s, t}) → o
  33. Circle(A, (0.25 * o)) → c_1
  34. Intersect(h, t) → I
  35. Intersect(h, t) → J
  36. Intersect(h, t) → K
  37. Intersect(c_1, s) → L,
  38. Intersect(c_1, h) → M,
  39. CircleSector(A, M, L) → d_1
  40. Dilate(A, If[2 < b ≤ 3, 3 - b, 0], F) → A'
  41. Intersect(l, r) → N
  42. Segment(C, N) → f_1
  43. Min({f_1, p, n, m}) → u
  44. Circle(C, (0.35 * u)) → e_1
  45. Circle(D, (0.35 * u)) → g_1
  46. Segment(A, A') → h_1
  47. Dilate(F, If[3 < b ≤ 4, 4 - b, 0], G) → F'
  48. Segment(F, F') → i_1
  49. Dilate(C, If[4 < b ≤ 5, 5 - b, 0], F) → C'
  50. Segment(C, C') → j_1
  51. Dilate(F, If[5 < b ≤ 6, 6 - b, 0], H) → F'_1
  52. Segment(F, F'_1) → k_1
  53. Intersect(e_1, j_1, 1) → O
  54. Intersect(e_1, j, 1) → P
  55. CircleSector(C, O, P) → p_1
  56. Intersect(g_1, p, 1) → Q
  57. Intersect(g_1, n, 1) → R
  58. CircleSector(D, Q, R) → q_1
  59. Rotate(d_1, If[6 < b ≤ 7, ((b - 6) * γ), γ], A) → r_1
  60. Dilate(D, If[7 < b ≤ 8, 8 - b, 0], F) → D'_1
  61. Dilate(D, If[7 < b ≤ 8, 8 - b, 0], H) → D'_2
  62. Segment(D, D'_1) → l_1
  63. Segment(D, D'_2) → m_1
  64. Dilate(H, If[8 < b ≤ 9, 9 - b, 0], G) → H'
  65. Segment(H, H') → n_1
  66. Line(C, n_1) → s_1
  67. Intersect(e_1, s_1, 2) → S
  68. Angle(O, C, S) → θ
  69. Dilate(C, If[10 < b ≤ 11, 11 - b, 0], H) → C'_1
  70. Vector(C, C'_1) → v
  71. Translate(p_1, v) → t_1
  72. Rotate(t_1, If[11 < b ≤ 12, ((b - 11) * θ), θ], H) → c_2
  73. Line(H, D) → b_1
  74. Dilate(D, If[12 < b ≤ 13, 13 - b, 0], H) → D'_3
  75. Vector(D, D'_3) → w
  76. Translate(q_1, w) → d_2
  77. Rotate(d_2, If[13 < b ≤ 14, ((-(b - 13)) * π), (-π)], H) → e_2