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Dynamic Illustration of Concyclic Relationship Between Two Circles and Their Com

Dynamic Illustration of Concyclic Relationship Between Two Circles and Their Com
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2017-06-30 09:45
CirclePolygonGeometric Construction
This example demonstrates the concyclic relationship between two circles and their common external tangents. Key mathematical concepts include circles, tangents, and concyclic points. The construction uses GeoGebra commands such as Circle, Tangent, Intersect, etc. Suitable for students at the junior high school level and above to learn about circle properties and concyclic points.

GeoGebra Commands & Parameters

  1. Circle(A, B) → c
  2. Segment(A, B) → f
  3. PointIn(a) → C
  4. Distance(C, c) → b
  5. Circle(C, d) → e
  6. Tangent(c, e) → g, h, i, j
  7. Intersect(c, h) → D
  8. Intersect(e, h) → G, L
  9. Intersect(e, g) → H, K
  10. Intersect(e, i) → I
  11. Intersect(c, i) → J, M
  12. Intersect(c, g) → O
  13. Circle(D, 2) → p
  14. Circle(O, 2) → r
  15. Circle(K, 2) → s
  16. Circle(I, 2) → t
  17. Circle(G, 2) → d_1
  18. Circle(J, 2) → e_1
  19. Intersect(e_1, i, 1) → P
  20. Intersect(e_1, i, 2) → Q
  21. Intersect(p, h, 1) → R
  22. Intersect(p, h, 2) → S
  23. Intersect(d_1, h, 1) → V
  24. Intersect(d_1, h, 2) → W
  25. Intersect(s, g, 2) → Z
  26. Intersect(s, g, 1) → A_1
  27. Intersect(r, g, 2) → B_1
  28. Intersect(r, g, 1) → C_1
  29. Dilate(A, If[0 < k ≤ 1, 1 - k, 0], D) → A'
  30. Dilate(A, If[0 < k ≤ 1, 1 - k, 0], M) → A'_1
  31. Dilate(A, If[0 < k ≤ 1, 1 - k, 0], O) → A'_3
  32. Angle(A, O, C_1) → α
  33. Angle(A, D, S) → δ
  34. Intersect(t, i, 1) → F_1
  35. Angle(C, I, F_1) → ζ
  36. Angle(C, G, W) → θ_1
  37. Dilate(C, If[0 < k ≤ 1, 1 - k, 0], G) → C'
  38. Dilate(C, If[0 < k ≤ 1, 1 - k, 0], I) → C'_2
  39. Dilate(C, If[0 < k ≤ 1, 1 - k, 0], K) → C'_3
  40. Intersect(i, j) → G_1
  41. Intersect(i, j) → H_1
  42. Line(C, H) → k_1
  43. Intersect(e, ee) → J_1, , ,
  44. Dilate(H, If[8 < k ≤ 9, 9 - k, 0], C) → H'_1
  45. Vector(H, H'_1) → w
  46. Dilate(H, If[11 < k ≤ 12, 12 - k, 0], C) → H'_2
  47. Vector(H, H'_2) → u_1
  48. Segment(A, B) → h_2
  49. Intersect(e, ee, 1) → W_1
  50. CircleArc(A, O, D) → k_2
  51. CircleArc(C, G, H) → p_2
  52. Point(k_2) → Z_1
  53. Point(p_2) → A_2
  54. Segment(A, Z_1) → i_2
  55. Segment(C, A_2) → j_2
  56. OrthogonalLine(Z_1, i_2) → l_2
  57. OrthogonalLine(A_2, j_2) → m_2
  58. Line(D, Z_1) → l
  59. Line(G, A_2) → m
  60. Intersect(h, m_2) → E
  61. Intersect(g, m_2) → F
  62. Intersect(l_2, h) → N
  63. Intersect(g, l_2) → T
  64. Intersect(m, l) → U
  65. Line(H, A_2) → n
  66. Line(O, Z_1) → q
  67. Intersect(n, q) → D_1
  68. Circle(A_2, D_1, Z_1) → c_1
  69. Segment(F, E) → f_1
  70. Segment(T, N) → g_1
  71. LineBisector(A_2, D_1) → h_1
  72. LineBisector(D_1, Z_1) → i_1
  73. Intersect(h_1, i_1) → E_1
  74. Dilate(D, If[0 < k ≤ 1, 1 - k, 0], U) → D'
  75. Segment(D, D') → j_1
  76. Dilate(G, If[0 < k ≤ 1, 1 - k, 0], U) → G'
  77. Segment(G, G') → l_1
  78. Dilate(H, If[1 < k ≤ 2, 2 - k, 0], D_1) → H'
  79. Dilate(O, If[1 < k ≤ 2, 2 - k, 0], D_1) → O'
  80. Segment(O, O') → m_1
  81. Segment(H, H') → n_1
  82. Dilate(E_1, If[3 < k ≤ 4, 4 - k, 0], U) → I_1
  83. Circle(E_1, I_1) → p_1