MathPlotAI Logo MathPlotAI

Circle Angle Relationships: Chord, Secant, Tangent

Circle Angle Relationships: Chord, Secant, Tangent
Loading the math board and drawing, please wait…
2014-05-14 05:45
CircleAngleGeometric Construction
In collections Circles
This example demonstrates the relationships between angles and arcs formed by chords, secants, and tangents in a circle. Key mathematical concepts include the Inscribed Angle Theorem, Central Angle Theorem, and Tangent-Secant Angle Theorem. The construction utilizes GeoGebra commands such as Circle, Line, OrthogonalLine, Intersect, Rotate, CircleArc, Point, Segment, Angle, CircleSector, and Dilate. It is suitable for high school mathematics education to help students visually understand the calculations and proofs of angles and arcs in circles.

GeoGebra Commands & Parameters

  1. Circle(A, 5) → c
  2. Line(A, a) → b
  3. OrthogonalLine(A, a) → d
  4. Intersect(c, d, 2) → B
  5. Rotate(B, 45*°, A) → B'
  6. Rotate(B, -45*°, A) → B'_1
  7. OrthogonalLine(B', a) → e
  8. OrthogonalLine(B'_1, a) → f
  9. Intersect(c, e, 1) → C
  10. Intersect(c, f, 1) → D
  11. CircleArc(A, B'_1, B') → g
  12. CircleArc(A, C, D) → h
  13. Point(g) → E
  14. Point(h) → F
  15. CircleArc(A, D, B'_1) → k
  16. Point(k) → G
  17. Point(k) → H
  18. Line(E, H) → i
  19. Line(G, F) → j
  20. Intersect(i, j) → P
  21. Segment(P, E) → l
  22. Segment(P, G) → m
  23. Segment(P, H) → n
  24. Segment(P, F) → p
  25. Angle(H, A, G) → Arc_2
  26. Angle(E, A, F) → Arc_1
  27. Rotate(E, (-Arc_2), A) → E'
  28. Angle(H, A, G) → γ
  29. Angle(E, P, F) → Angle
  30. CircleArc(A, E, F) → q
  31. CircleArc(A, E', E) → r
  32. CircleArc(A, E, E') → s
  33. CircleSector(A, E, F) → t
  34. Dilate(A, 0.05, E) → A'
  35. Rotate(A', (-Arc_2), A) → A''
  36. CircleArc(A, E, E') → c_1
  37. CircleArc(A, E, A'') → d_1
  38. CircleArc(A, A', A'') → e_1
  39. Dilate(A, 0.05, F) → A'_1
  40. Rotate(A'_1, (-Angle), A) → L
  41. Rotate(L, (-Angle), A) → L'
  42. CircleArc(A, H, G) → f_1
  43. CircleArc(A, G, H) → g_1
  44. Circle(P, Segment[A, A'_1]) → h_1
  45. Intersect(h_1, p, 1) → M
  46. Intersect(h_1, l, 1) → N
  47. CircleArc(P, N, M) → k_1
  48. CircleArc(A, L, F) → p_1
  49. CircleArc(A, L', L) → q_1
  50. Segment(L, A) → a_1
  51. Segment(F, A) → b_1
  52. Segment(L', A) → i_1
  53. Segment(E, A) → j_1
  54. Angle(L', A, L) → ε
  55. Angle(L, A, F) → ζ
  56. Segment(A, F) → l_1
  57. Circle(A_1, 5) → c_2
  58. Line(A_1, a_2) → b_3
  59. OrthogonalLine(A_1, a_2) → d_2
  60. Intersect(c_2, d_2, 2) → B_1
  61. Rotate(B_1, 45*°, A_1) → B'_2
  62. Rotate(B_1, -45*°, A_1) → B'_3
  63. OrthogonalLine(B'_2, a_2) → e_2
  64. OrthogonalLine(B'_3, a_2) → f_2
  65. Intersect(c_2, e_2, 1) → C_1
  66. Intersect(c_2, f_2, 1) → D_1
  67. CircleArc(A_1, B'_3, B'_2) → g_2
  68. CircleArc(A_1, C_1, D_1) → h_2
  69. Point(g_2) → E_1
  70. Point(h_2) → F_1
  71. CircleArc(A_1, D_1, B'_3) → k_2
  72. Angle(E_1, A_1, F_1) → β_1
  73. CircleArc(A_1, E_1, F_1) → q_2
  74. CircleSector(A_1, E_1, F_1) → t_1
  75. Segment(F_1, A_1) → b_2
  76. Segment(E_1, A_1) → j_2
  77. Segment(A_1, F_1) → l_2