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Dynamic Relationships in a Cyclic Quadrilateral

Dynamic Relationships in a Cyclic Quadrilateral
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2017-09-25 05:36
CirclePolygonGeometric ConstructionLocus
This example demonstrates dynamic and modifiable relationships within a cyclic quadrilateral, involving concepts like circles, rotations, dilations, and vectors. It uses commands such as Circle, Point, Rotate, CircleArc, Segment, Intersect, Dilate, Vector, Translate, Angle, Ray, Polygon, Line, OrthogonalLine, and LineBisector. Ideal for high school students and educators exploring geometric transformations and dynamic geometry.

GeoGebra Commands & Parameters

  1. Circle(A_1, B_1) → c
  2. Point(c) → C_1
  3. Rotate(C_1, 2°, A_1) → C'
  4. CircleArc(A_1, C', C_1) → d
  5. Point(d) → D
  6. CircleArc(A_1, D, C_1) → e
  7. Point(e) → C
  8. CircleArc(A_1, C, C_1) → f
  9. Point(f) → B
  10. CircleArc(A_1, B, C_1) → g
  11. Point(g) → A
  12. Segment(D, C) → b
  13. Segment(C, B) → h
  14. Segment(D, A) → a
  15. Segment(A, B) → i
  16. Circle(B, (0.2 * Distance[A_1, B_1])) → k
  17. Circle(C, (0.2 * Distance[A_1, B_1])) → p
  18. Circle(A, (0.2 * Distance[A_1, B_1])) → q
  19. Circle(D, (0.2 * Distance[A_1, B_1])) → r
  20. Intersect(k, i, 1) → H_1
  21. Intersect(k, h, 1) → I
  22. Intersect(p, h, 1) → J
  23. Intersect(p, b, 1) → K
  24. Intersect(r, a, 1) → L
  25. Intersect(r, b, 1) → M
  26. Intersect(q, a, 1) → N
  27. Intersect(q, i, 1) → O
  28. Circle(A_1, B_1) → e_1
  29. Dilate(C, If[0 < j ≤ 1, 1 - j, 0], A) → E'
  30. Vector(C, E') → u
  31. Translate(J, u) → J'
  32. Angle(O, A, J') → α
  33. Ray(D, A) → l
  34. Point(l) → P
  35. Dilate(B, If[2 < j ≤ 3, 3 - j, 0], D) → F'
  36. Vector(B, F') → v
  37. Ray(A, D) → n
  38. Point(n) → Q
  39. Translate(I, v) → I'
  40. Angle(I', D, M) → β
  41. Polygon(A, D, C, B) → poly1, g_2, d_2, e_2, f_2
  42. Line(C, D) → i_1
  43. OrthogonalLine(A, i_1) → j_1
  44. OrthogonalLine(B, i_1) → l_1
  45. Intersect(i_1, j_1) → E
  46. Intersect(i_1, l_1) → F
  47. Line(B, A) → m_1
  48. OrthogonalLine(D, m_1) → n_1
  49. OrthogonalLine(C, m_1) → p_1
  50. Intersect(p_1, m_1) → G
  51. Intersect(n_1, m_1) → H
  52. Circle(H, G, F) → q_1
  53. Dilate(A, If[0 < j ≤ 1, 1 - j, 0], E) → A'
  54. Dilate(D, If[0 < j ≤ 1, 1 - j, 0], E) → D'
  55. Segment(D, D') → s
  56. Segment(A, A') → t
  57. Angle(C, E, A) → γ
  58. Dilate(B, If[1 < j ≤ 2, 2 - j, 0], F) → B'
  59. Dilate(C, If[1 < j ≤ 2, 2 - j, 0], F) → C'_1
  60. Segment(C, C'_1) → f_1
  61. Segment(B, B') → g_1
  62. Angle(B, F, D) → δ
  63. Dilate(C, If[2 < j ≤ 3, 3 - j, 0], G) → C'_2
  64. Dilate(B, If[2 < j ≤ 3, 3 - j, 0], G) → B'_1
  65. Segment(C, C'_2) → m
  66. Segment(B, B'_1) → h_1
  67. Angle(A, G, C) → ε
  68. Dilate(D, If[3 < j ≤ 4, 4 - j, 0], H) → D'_1
  69. Dilate(A, If[3 < j ≤ 4, 4 - j, 0], H) → A'_1
  70. Segment(D, D'_1) → k_1
  71. Segment(A, A'_1) → r_1
  72. Angle(D, H, B) → ζ
  73. LineBisector(G, F) → s_1
  74. LineBisector(E, H) → t_1
  75. Intersect(s_1, t_1) → R
  76. Dilate(R, If[5 < j ≤ 6, 6 - j, 0], G) → R'
  77. Circle(R, R') → c_1
  78. Segment(R, R') → a_1
  79. Circle(C, U, B) → d_1
  80. Circle(A, V, D) → c_2
  81. LineBisector(C, U) → b_1
  82. LineBisector(U, B) → h_2
  83. Intersect(b_1, h_2) → W
  84. LineBisector(V, D) → i_2
  85. LineBisector(V, A) → j_2
  86. Intersect(j_2, i_2) → Z
  87. Dilate(C, If[8 < j ≤ 9, 9 - j, 0], W) → C'_3
  88. Dilate(A, If[8 < j ≤ 9, 9 - j, 0], Z) → A'_2
  89. Segment(C, C'_3) → k_2
  90. Segment(A, A'_2) → l_2
  91. Rotate(C, If[9 < j ≤ 10, ((-(j - 9)) * π), (-π)], W) → C'_4
  92. Rotate(C'_3, If[9 < j ≤ 10, ((-(j - 9)) * π), (-π)], W) → D_1
  93. Segment(C'_4, D_1) → m_2
  94. Rotate(A, If[9 < j ≤ 10, ((-(j - 9)) * π), (-π)], Z) → A'_3
  95. Rotate(A'_2, If[9 < j ≤ 10, ((-(j - 9)) * π), (-π)], Z) → E_1
  96. Segment(A'_3, E_1) → n_2
  97. Dilate(W, If[10 < j ≤ 11, 11 - j, 0], U) → W'
  98. Dilate(Z, If[11 < j ≤ 12, 12 - j, 0], T) → Z'
  99. Circle(W, W') → p_2
  100. Segment(W, W') → q_2
  101. Segment(Z, Z') → r_2
  102. Circle(Z, Z') → s_2