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Angles Formed by Secants and Tangents

Angles Formed by Secants and Tangents
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2016-04-16 05:16
CircleAngleTransformationGeometric Construction
In collections Geometry Review Circles (Theorems)
This GeoGebra example demonstrates angles formed outside a circle by two secants, a secant and a tangent, or two tangents. It explores circle geometry, intercepted arcs, angle measures, and relationships among secants and tangents. The construction uses Circle, Semicircle, PointIn, Intersect, Segment, Line, CircleArc, CircleSector, Angle, Rotate, Translate, Dilate, Vector, and Min to create points, arcs, segments, tangent and secant lines, angle labels, and dynamic transformations. It is suitable for middle school or high school geometry discovery activities, teacher demonstrations, and student investigations of secant-tangent angle theorems.

GeoGebra Commands & Parameters

  1. Circle(A, B) → c
  2. PointIn(a) → C
  3. Semicircle(A, C) → d
  4. Intersect(c, d, 1) → D
  5. Semicircle(C, A) → e
  6. Intersect(c, e, 1) → E
  7. CircleArc(A, D, E) → f
  8. CircleArc(A, E, D) → g
  9. Point(f) → F
  10. CircleArc(A, F, E) → h
  11. Point(h) → G
  12. Segment(C, F) → i
  13. Segment(C, G) → j
  14. Angle(F, A, G) → α
  15. Rotate(F, α / 2, A) → F'
  16. Angle(F, A, F') → γ
  17. Intersect(g, i, 1) → H
  18. Intersect(g, j, 1) → I
  19. Min({Distance[A, B], Distance[C, H], Distance[C, I]}) → k
  20. Circle(A, (0.4 * k)) → p
  21. Segment(A, F) → l
  22. Segment(A, G) → m
  23. Intersect(p, l, 1) → J
  24. Segment(A, I) → n
  25. Segment(A, H) → q
  26. Intersect(p, n, 1) → K
  27. Intersect(p, q, 1) → L
  28. Segment(A, F') → r
  29. Angle(I, A, H) → β
  30. Dilate(A, If[0 < b ≤ 1, 1 - b, 0], F) → A'
  31. Dilate(A, If[0 < b ≤ 1, 1 - b, 0], G) → A'_1
  32. Segment(A, A') → s
  33. Segment(A, A'_1) → t
  34. Dilate(A, If[0 < b ≤ 1, 1 - b, 0], H) → A'_2
  35. Dilate(A, If[0 < b ≤ 1, 1 - b, 0], I) → A'_3
  36. Segment(A, A'_2) → f_1
  37. Segment(A, A'_3) → g_1
  38. Dilate(A, If[1 < b ≤ 2, 2 - b, 0], F') → A'_4
  39. Segment(A, A'_4) → h_1
  40. Rotate(J, If[1 < b ≤ 2, ((b - 1) * γ), γ], A) → J'
  41. CircleSector(A, J, J') → c_1
  42. Rotate(I, β / 2, A) → I'
  43. Dilate(A, If[2 < b ≤ 3, 3 - b, 0], I') → A'_5
  44. Segment(A, A'_5) → i_1
  45. Rotate(K, If[2 < b ≤ 3, ((b - 2) * β / 2), β / 2], A) → K'
  46. CircleSector(A, K, K') → d_1
  47. Rotate(c_1, If[3 < b ≤ 4, ((b - 3) * γ), γ], A) → e_1
  48. Rotate(d_1, If[4 < b ≤ 5, ((b - 4) * β / 2), β / 2], A) → k_1
  49. Segment(A, F') → j_1
  50. Intersect(p, j_1) → M,
  51. Segment(A, I') → l_1
  52. Intersect(p, l_1) → N,
  53. Angle(L, A, M) → δ
  54. Rotate(A, If[6 < b ≤ 7, ((b - 6) * δ), δ], A) → A'_6
  55. Rotate(A'_5, If[7 < b ≤ 8, ((b - 7) * δ), δ], A) → O
  56. Segment(A'_6, O) → m_1
  57. Rotate(k_1, If[7 < b ≤ 8, ((b - 7) * δ), δ], A) → p_1
  58. Angle(F, C, G) → ε
  59. Rotate(J, If[8 < b ≤ 9, ((b - 8) * ε), ε], A) → J'_1
  60. CircleSector(A, J, J'_1) → q_1
  61. Dilate(A, If[10 < b ≤ 11, 11 - b, 0], C) → A'_7
  62. Vector(A, A'_7) → u
  63. Translate(q_1, u) → r_1
  64. Line(C, s) → n_1
  65. Point(n_1) → P
  66. Angle(P, C, H) → ζ
  67. Rotate(r_1, If[11 < b ≤ 12, ((b - 11) * ζ), ζ], C) → s_1
  68. Angle(A, D, C) → η
  69. Angle(C, E, A) → θ
  70. CircleArc(A, F, G) → t_1
  71. CircleArc(A, I, H) → c_2
  72. CircleArc(A, H, F) → d_2
  73. CircleArc(A, G, I) → e_2

AI Drawing Prompt Reference

Please draw a diagram that demonstrates angles formed outside a circle by two secants, a secant and a tangent, or two tangents. The diagram should include a circle, several points, arcs, segments, tangent and secant lines, angle labels, and dynamic transformations. First, draw a circle, then select several points inside the circle and connect these points to form secants and tangents. Calculate and label the angles formed by these line segments. Finally, perform rotations, translations, and dilations to show the relationships among the angles.
The prompt above was compiled from the original drawing commands and is for reference only.