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Dynamic Relationship Between Circle and Cyclic Quadrilateral

Dynamic Relationship Between Circle and Cyclic Quadrilateral
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2017-04-15 10:35
CirclePolygonAngleTransformationGeometric Construction
In collections Circles (Theorems)
This example demonstrates the dynamic relationship between a circle and a cyclic quadrilateral. Through transformations such as rotation, translation, and dilation, the position of the quadrilateral on the circle is shown to change. Key mathematical concepts include circles, quadrilaterals, and angles. GeoGebra commands used include Circle, Point, Rotate, Intersect, and Dilate. Suitable for students at the middle school level and above for learning geometric transformations and dynamic geometry.

GeoGebra Commands & Parameters

  1. Circle(A, B) → c
  2. Point(c) → C
  3. Rotate(C, 2°, A) → C'
  4. CircleArc(A, C', C) → d
  5. Point(d) → D
  6. CircleArc(A, D, C) → e
  7. Point(e) → E
  8. CircleArc(A, E, C) → f
  9. Point(f) → F
  10. CircleArc(A, F, C) → g
  11. Point(g) → G
  12. Segment(D, E) → b
  13. Segment(E, F) → h
  14. Segment(D, G) → a
  15. Segment(G, F) → i
  16. Circle(F, (0.2 * Distance[A, B])) → k
  17. Circle(E, (0.2 * Distance[A, B])) → p
  18. Circle(G, (0.2 * Distance[A, B])) → q
  19. Circle(D, (0.2 * Distance[A, B])) → r
  20. Intersect(k, i, 1) → H
  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, B) → e_1
  29. Dilate(E, If[0 < j ≤ 1, 1 - j, 0], G) → E'
  30. Vector(E, E') → u
  31. Translate(J, u) → J'
  32. Angle(O, G, J') → α
  33. Ray(D, G) → l
  34. Point(l) → P
  35. Dilate(F, If[2 < j ≤ 3, 3 - j, 0], D) → F'
  36. Vector(F, F') → v
  37. Ray(G, D) → n
  38. Point(n) → Q
  39. Translate(I, v) → I'
  40. Angle(I', D, M) → β
  41. Polygon(G, D, E, F) → poly1, g_2, d_2, e_2, f_2
  42. Ray(F, D) → i_1
  43. Ray(G, E) → j_1
  44. Point(i_1) → R
  45. OrthogonalLine(R, i_1) → l_1
  46. OrthogonalLine(G, j_1) → m_1
  47. Intersect(e_1, m_1) → S, T
  48. Intersect(e_1, l_1) → U, V
  49. Intersect(l_1, j_1) → W
  50. Intersect(i_1, m_1) → Z
  51. Segment(F, W) → n_1
  52. Point(n_1) → A_1
  53. Segment(A_1, Z) → p_1
  54. Segment(W, G) → m
  55. Segment(G, Z) → s
  56. Segment(V, S) → t
  57. Segment(W, Z) → f_1
  58. Line(E, D) → g_1
  59. Line(F, W) → h_1
  60. Intersect(h_1, g_1) → B_1
  61. Line(G, Z) → k_1
  62. Intersect(g_1, k_1) → C_1
  63. Line(V, S) → q_1
  64. Line(W, Z) → r_1
  65. Circle(V, 2) → c_1
  66. Circle(B_1, 2) → d_1
  67. Circle(W, 2) → s_1
  68. Intersect(s_1, r_1, 1) → D_1
  69. Intersect(d_1, g_1, 1) → E_1
  70. Intersect(c_1, q_1, 1) → F_1
  71. Ray(W, D_1) → t_1
  72. Ray(B_1, E_1) → b_1
  73. Ray(V, F_1) → h_2
  74. Segment(V, B_1) → i_2
  75. Segment(B_1, W) → j_2
  76. Min(i_2, j_2) → o
  77. Circle(V, (0.35 * o)) → c_2
  78. Intersect(c_2, h_2, 1) → G_1
  79. Line(W, V) → k_2
  80. Intersect(c_2, k_2, 1) → H_1
  81. Dilate(F, If[0 < j ≤ 1, 1 - j, 0], V) → F'_1
  82. Segment(F, F'_1) → l_2
  83. Dilate(V, If[1 < j ≤ 2, 2 - j, 0], W) → V'
  84. Segment(V, V') → m_2
  85. Dilate(F, If[0 < j ≤ 1, 1 - j, 0], D) → E'_1
  86. Segment(F, E'_1) → n_2
  87. Angle(D, F, V) → γ
  88. Dilate(D, If[1 < j ≤ 2, 2 - j, 0], Z) → D'
  89. Segment(D, D') → p_2
  90. Angle(Z, G, W) → δ
  91. Dilate(G, If[2 < j ≤ 3, 3 - j, 0], S) → G'
  92. Segment(G, G') → q_2
  93. Dilate(G, If[2 < j ≤ 3, 3 - j, 0], E) → G'_1
  94. Segment(G, G'_1) → r_2
  95. Intersect(c_2, k_2) → I_1, J_1
  96. Dilate(E, If[3 < j ≤ 4, 4 - j, 0], W) → E'_2
  97. Dilate(S, If[3 < j ≤ 4, 4 - j, 0], Z) → S'
  98. Segment(E, E'_2) → s_2
  99. Segment(S, S') → t_2
  100. Dilate(S, If[4 < j ≤ 5, 5 - j, 0], V) → S'_1
  101. Segment(S, S'_1) → a_1
  102. Dilate(D, If[5 < j ≤ 6, 6 - j, 0], C_1) → D'_1
  103. Dilate(E, If[5 < j ≤ 6, 6 - j, 0], B_1) → E'_3
  104. Dilate(Z, If[6 < j ≤ 7, 7 - j, 0], W) → Z'
  105. Segment(Z, Z') → a_2
  106. Segment(D, D'_1) → b_2
  107. Segment(E, E'_3) → f_3
  108. Angle(F_1, V, J_1) → ε
  109. Rotate(J_1, If[7 < j ≤ 8, ((-(j - 7)) * ε), (-ε)], V) → K_1
  110. CircleSector(V, G_1, J_1) → c_3
  111. CircleSector(V, K_1, J_1) → d_3
  112. Dilate(V, If[8.5 < j ≤ 9.5, 9.5 - j, 0], B_1) → V'_1
  113. Dilate(B_1, If[9.5 < j ≤ 10.5, 10.5 - j, 0], W) → L_1
  114. Vector(V, V'_1) → w
  115. Translate(d_3, w) → e_3
  116. Vector(B_1, L_1) → u_1
  117. Translate(e_3, u_1) → g_3