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Cyclic Quadrilateral: Proof Hint

Cyclic Quadrilateral: Proof Hint
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2018-06-12 20:46
CirclePolygonAngleGeometric ConstructionSlider Animation
This example demonstrates the proof hint for a cyclic quadrilateral, involving the corollary of the inscribed angle theorem. It is dynamic and modifiable, allowing observation of the properties of cyclic quadrilaterals. Mainly using Circle, Point, Polygon, Segment, Angle, Rotate commands in GeoGebra. Suitable for students at junior high school level and above for geometry proof learning.

GeoGebra Commands & Parameters

  1. Circle(A, B) → c
  2. Point(c) → C
  3. Point(c) → D
  4. CircleArc(A, C, D) → d
  5. CircleArc(A, D, C) → e
  6. Point(e) → E
  7. CircleArc(A, D, E) → f
  8. CircleArc(A, E, C) → g
  9. Point(g) → F
  10. Polygon(F, C, D, E) → q1, f_1, c_1, d_1, e_1
  11. Segment(C, D) → h
  12. Segment(D, E) → i
  13. Segment(E, F) → j
  14. Ray(F, C) → k
  15. Min({i, j, h, Distance[F, C]}) → a
  16. Circle(E, (0.2 * a)) → p
  17. Intersect(p, j) → G,
  18. Angle(F, E, D) → α
  19. Rotate(G, If[0 < b ≤ 1, (b * α), α], E) → G'
  20. CircleSector(E, G, G') → q
  21. Dilate(E, If[1.5 < b ≤ 2.5, 2.5 - b, 0], C) → E'
  22. Vector(E, E') → u
  23. Translate(q, u) → q'
  24. Line(E, k) → l
  25. Point(l) → H
  26. Angle(G, E, H) → β
  27. Circle(C, (0.2 * a)) → r
  28. Intersect(r, h, 1) → I
  29. Rotate(I, If[0 < b ≤ 1, (b * (π - α)), π - α], C) → I'
  30. CircleSector(C, I, I') → s
  31. Dilate(A, If[1 < b ≤ 2, 2 - b, 0], D) → A'
  32. Dilate(A, If[1 < b ≤ 2, 2 - b, 0], F) → A'_1
  33. Segment(A, A') → m
  34. Segment(A, A'_1) → n
  35. Line(A, i) → t
  36. Angle(F, A, D) → γ
  37. Rotate(F, γ / 2, A) → F'
  38. Angle(F, A, F') → δ
  39. Dilate(E, If[2 < b ≤ 3, 3 - b, 0], A) → E'_1
  40. Vector(E, E'_1) → v
  41. Translate(q, v) → q'_1
  42. Point(t) → J
  43. Angle(J, A, D) → ε
  44. Rotate(q'_1, If[2 < b ≤ 3, ((b - 2) * ε), ε], E'_1) → g_1
  45. Dilate(C, If[3 < b ≤ 4, 4 - b, 0], A) → C'
  46. Vector(C, C') → w
  47. Translate(s, w) → s'
  48. Line(A, h) → h_1
  49. Point(h_1) → K
  50. Angle(D, A, K) → ζ
  51. Rotate(s', If[3 < b ≤ 4, ((b - 3) * ((2 * π) - ζ)), (2 * π) - ζ], C') → s''
  52. Dilate(A, If[4 < b ≤ 5, 5 - b, 0], F') → A'_2
  53. Rotate(D, π - γ / 2, A) → D'
  54. Dilate(A, If[4 < b ≤ 5, 5 - b, 0], D') → A'_3
  55. Segment(A, A'_2) → i_1
  56. Segment(A, A'_3) → j_1
  57. Rotate(I, If[4 < b ≤ 5, ((b - 4) * (π - α)), π - α], C) → I'_1
  58. Rotate(G, If[4 < b ≤ 5, ((b - 4) * α), α], E) → G'_1
  59. CircleSector(C, I, I'_1) → k_1
  60. CircleSector(E, G, G'_1) → p_1
  61. Dilate(E, If[5.5 < b ≤ 6.5, 6.5 - b, 0], A) → E'_2
  62. Vector(E, E'_2) → u_1
  63. Translate(p_1, u_1) → q_1
  64. Line(A, j) → l_1
  65. Point(l_1) → L
  66. Angle(F, A, L) → η
  67. Rotate(q_1, If[5.5 < b ≤ 6.5, ((b - 5.5) * ((2 * π) - η)), (2 * π) - η], E'_2) → r_1
  68. CircleArc(A, F, D) → s_1
  69. CircleArc(A, D, F) → t_1
  70. CircleArc(A, F, D) → c_2
  71. Dilate(C, If[6.5 < b ≤ 7.5, 7.5 - b, 0], A) → C'_1
  72. Vector(C, C'_1) → v_1
  73. Translate(k_1, v_1) → d_2
  74. Line(A, k) → m_1
  75. Point(m_1) → M
  76. Angle(M, A, F) → θ
  77. Rotate(d_2, If[6.5 < b ≤ 7.5, ((b - 6.5) * θ), θ], C'_1) → e_2
  78. CircleArc(A, D, F) → f_2
  79. Segment(F, C) → n_1

AI Drawing Prompt Reference

Draw a circle and select two points on it. Use these points as centers to draw two tangent circles. The intersection points form a quadrilateral. Label the interior angles and demonstrate the sum of interior angles theorem. Add a slider to dynamically adjust the figure and observe the angle changes.
The prompt above was compiled from the original drawing commands and is for reference only.