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60-Degree Sector Surprise!

60-Degree Sector Surprise!
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2017-05-28 09:32
CircleTrigonometryGeometric Construction
This dynamic illustration demonstrates a special phenomenon found inside any 60-degree sector of a circle using GeoGebra commands such as rotation, mirroring, and circle arcs. Suitable for high school geometry and trigonometry studies.

GeoGebra Commands & Parameters

  1. Ray(A, B) → j
  2. Point(j) → D
  3. Segment(A, D) → r_1
  4. Line(A, B) → f
  5. Rotate(B, π / 3, A) → B'
  6. Angle(B, A, B') → α
  7. Rotate(B', π / 3, A) → B''
  8. Angle(B', A, B'') → β
  9. Ray(A, B'') → g
  10. Ray(A, B') → h
  11. Point(h) → C
  12. Mirror(B, A) → B'_1
  13. Segment(C, D) → i
  14. Ray(A, B'_1) → k
  15. Segment(A, D) → l
  16. Segment(A, C) → m
  17. Circle(A, (0.15 * r_1)) → c
  18. Intersect(c, k, 1) → E
  19. Intersect(c, g) → F,
  20. CircleSector(A, F, E) → d
  21. Rotate(d, If[0 < a ≤ 1, ((-a) * π / 3), ((-π)) / 3], A) → d'
  22. Rotate(d', If[2 < a ≤ 3, ((-(a - 2)) * π / 3), ((-π)) / 3], A) → d''
  23. Line(A, C) → n
  24. OrthogonalLine(D, n) → p
  25. Intersect(n, p) → J
  26. Line(A, D) → q
  27. OrthogonalLine(C, q) → r
  28. Intersect(r, q) → H
  29. Line(A, C) → n_1
  30. Circle(C, 1) → e_1
  31. Intersect(e_1, n_1, 2) → N
  32. Circle(A, 1) → p_1
  33. Intersect(p_1, n_1) → O, P
  34. Ray(A, D) → s_1
  35. Circle(D, 1) → t_1
  36. Intersect(t_1, s_1) → Q, R
  37. Line(A, C) → q_1
  38. Circle(A, D) → c_1
  39. Intersect(c_1, q_1) → U, V
  40. CircleArc(A, D, V) → e
  41. Point(e) → G
  42. Ray(D, B) → s
  43. Midpoint(A, D) → I
  44. Dilate(A, If[3 < a ≤ 4, 4 - a, 0], I) → A'
  45. Circle(V, 1) → t
  46. Midpoint(A, V) → K
  47. Midpoint(G, V) → L
  48. Midpoint(D, G) → S
  49. Distance(I, S) → distanceIS
  50. Distance(S, L) → distanceSL
  51. Distance(L, K) → distanceLK
  52. Distance(K, I) → distanceKI
  53. Intersect(t, q_1, 2) → T
  54. Ray(V, T) → b
  55. Dilate(A, If[3 < a ≤ 4, 4 - a, 0], K) → A'_1
  56. Segment(A, A') → f_1
  57. Segment(A, A'_1) → g_1
  58. Rotate(A, If[4 < a ≤ 5, ((-(a - 4)) * π), (-π)], I) → A'_2
  59. Rotate(A', If[4 < a ≤ 5, ((-(a - 4)) * π), (-π)], I) → A''
  60. Segment(A'_2, A'') → h_1
  61. Rotate(A, If[4 < a ≤ 5, ((-(a - 4)) * π), (-π)], K) → A'_3
  62. Rotate(A'_1, If[4 < a ≤ 5, ((-(a - 4)) * π), (-π)], K) → W
  63. Segment(A'_3, W) → i_1
  64. Dilate(G, If[5 < a ≤ 6, 6 - a, 0], S) → G'
  65. Dilate(G, If[5 < a ≤ 6, 6 - a, 0], L) → G'_1
  66. Segment(G, G') → a_1
  67. Segment(G, G'_1) → b_1
  68. Dilate(L, If[6 < a ≤ 7, 7 - a, 0], V) → L'
  69. Dilate(S, If[6 < a ≤ 7, 7 - a, 0], D) → S'
  70. Segment(L, L') → d_1
  71. Segment(S, S') → f_2
  72. Rotate(G, If[7 < a ≤ 8, ((a - 7) * π), π], S) → G'_2
  73. Rotate(G', If[7 < a ≤ 8, ((a - 7) * π), π], S) → G''
  74. Segment(G'_2, G'') → j_1
  75. Rotate(G, If[7 < a ≤ 8, ((a - 7) * π), π], L) → G'_3
  76. Rotate(G'_1, If[7 < a ≤ 8, ((a - 7) * π), π], L) → Z
  77. Segment(G'_3, Z) → k_1
  78. Dilate(I, If[8 < a ≤ 9, 9 - a, 0], S) → I'
  79. Segment(I, I') → l_1
  80. Angle(S, I, K) → γ
  81. Rotate(I, If[9 < a ≤ 10, ((a - 9) * γ), γ], I) → I'_1
  82. Rotate(I', If[9 < a ≤ 10, ((a - 9) * γ), γ], I) → I''
  83. Segment(I'_1, I'') → m_1
  84. Dilate(I, If[10 < a ≤ 11, 11 - a, 0], K) → I'_2
  85. Vector(I, I'_2) → u
  86. Translate(I, u) → I'_3
  87. Translate(I', u) → I''_1
  88. Segment(I'_3, I''_1) → g_2
  89. Dilate(K, If[11 < a ≤ 12, 12 - a, 0], L) → K'
  90. Vector(K, K') → v
  91. Translate(I'_1, v) → M
  92. Translate(I'', v) → I'''
  93. Segment(M, I''') → h_2
  94. CircleSector(A, D, V) → c_2
  95. Polygon(I, S, L, K) → poly1, i_2, s_2, l_2, k_2