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Angle Formed by a Secant & Tangent: What Do You See?

Angle Formed by a Secant & Tangent: What Do You See?
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2015-11-22 09:53
CircleAngleGeometric Construction
This example demonstrates the angle formed by a secant and a tangent. Key mathematical concepts include the properties of circles, isosceles triangles, and angle relationships. Using GeoGebra commands such as Circle, Point, Segment, Tangent, etc., dynamic exploration of these geometric elements is possible. Suitable for middle school and above students or teachers to understand circle properties and geometric construction techniques.

GeoGebra Commands & Parameters

  1. Circle(A, B) → c
  2. Point(c) → C
  3. Segment(C, A) → a
  4. OrthogonalLine(C, a) → b
  5. Point(b) → E
  6. Tangent(E, c) → l, m
  7. Intersect(c, m) → K
  8. CircleArc(A, C, K) → t
  9. Point(t) → D
  10. Ray(E, D) → d
  11. Ray(E, C) → e
  12. Segment(A, D) → f
  13. Angle(C, A, D) → α
  14. Rotate(C, α / 2, A) → C'
  15. Angle(C, A, C') → β
  16. Circle(A, 1.5) → h
  17. Intersect(h, a, 1) → H
  18. Intersect(h, f, 1) → I
  19. Angle(I, A, H) → γ
  20. CircleArc(A, K, C) → s
  21. Ray(E, D) → n
  22. Intersect(s, n, 1) → F
  23. Segment(A, F) → q
  24. Intersect(h, q, 1) → L
  25. CircleArc(A, F, C) → r
  26. CircleArc(A, D, F) → c_1
  27. CircleArc(A, C, D) → d_1
  28. Angle(F, A, C) → δ
  29. Rotate(F, δ / 2, A) → F'
  30. Angle(F, A, F') → ε
  31. Vector(A, E) → u
  32. Translate(H, u) → H'
  33. Dilate(A, If[1.5 < j < 2.5, 2.5 - j, 0], E) → A'
  34. Vector(A, A') → v
  35. Line(A, n) → a_1
  36. Intersect(h, a_1) → M, N
  37. Angle(a, e) → λ
  38. Ray(E, K) → b_1
  39. Angle(E, D, A) → μ
  40. CircleArc(A, C, K) → t_1
  41. Point(t_1) → O
  42. CircleArc(A, K, C) → c_2
  43. Point(c_2) → P
  44. Segment(O, C) → i_1
  45. Segment(O, F) → j_1
  46. Segment(P, D) → l_1
  47. Segment(P, C) → m_1
  48. Circle(O, 1.5) → g
  49. Circle(P, 1.5) → k
  50. Intersect(g, j_1, 1) → G
  51. Intersect(g, i_1, 1) → J
  52. Intersect(k, l_1, 1) → Q
  53. Intersect(k, m_1, 1) → R
  54. CircleSector(O, G, J) → p
  55. CircleSector(P, R, Q) → e_1
  56. Dilate(O, If[0 < j < 1, 1 - j, 0], P) → O'
  57. Vector(O, O') → w
  58. Translate(p, w) → p'
  59. Line(O, l_1) → i
  60. Intersect(g, i, 2) → S
  61. Angle(J, O, S) → ζ
  62. Rotate(p', If[0 < j < 1, (j * ζ), 0], O') → p''
  63. Vector(O, P) → u_1
  64. Translate(p, u_1) → p'_1
  65. Rotate(p'_1, ζ, P) → f_1
  66. Translate(G, u_1) → G'
  67. Rotate(G', ζ, P) → G''
  68. CircleSector(P, R, G'') → g_1
  69. Dilate(P, If[1.5 < j < 2.5, 2.5 - j, 0], E) → P'
  70. Vector(P, P') → v_1
  71. Translate(g_1, v_1) → h_1
  72. Vector(P, E) → w_1
  73. Translate(G'', w_1) → G'''
  74. Circle(E, 1.5) → k_1
  75. Intersect(k_1, n, 1) → T
  76. Angle(G''', E, T) → η
  77. Translate(g_1, w_1) → p_1
  78. Rotate(p_1, η, E) → q_1
  79. Rotate(h_1, If[1.5 < j < 2.5, ((j - 1.5) * η), 0], P') → r_1