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Dynamic Relationship Between Angles in Tangent Circles

Dynamic Relationship Between Angles in Tangent Circles
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2017-06-27 07:33
CircleAngleVectorGeometric Construction
This example dynamically illustrates the relationship between two angles formed within the context of common internal and external tangents to circles. It involves core math concepts like circles, tangent lines, angles, vectors, and geometric transformations. Constructed using GeoGebra commands such as Circle, Segment, PointIn, Distance, Tangent, Intersect, Dilate, Angle, Line, Rotate, etc., it is suitable for high school and above students to explore geometric relationships and dynamic constructions.

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(c, j) → E, N
  9. Intersect(e, j) → F
  10. Intersect(e, h) → G, L
  11. Intersect(e, g) → H, K
  12. Intersect(e, i) → I
  13. Intersect(c, i) → J, M
  14. Intersect(c, g) → O
  15. Circle(D, 2) → p
  16. Circle(E, 2) → q
  17. Circle(O, 2) → r
  18. Circle(K, 2) → s
  19. Circle(I, 2) → t
  20. Circle(F, 2) → c_1
  21. Circle(G, 2) → d_1
  22. Circle(J, 2) → e_1
  23. Intersect(e_1, i, 1) → P
  24. Intersect(e_1, i, 2) → Q
  25. Intersect(p, h, 1) → R
  26. Intersect(p, h, 2) → S
  27. Intersect(c_1, j, 2) → T
  28. Intersect(c_1, j, 1) → U
  29. Intersect(d_1, h, 1) → V
  30. Intersect(d_1, h, 2) → W
  31. Intersect(s, g, 2) → Z
  32. Intersect(s, g, 1) → A_1
  33. Intersect(r, g, 2) → B_1
  34. Intersect(r, g, 1) → C_1
  35. Intersect(q, j, 2) → D_1
  36. Intersect(q, j, 1) → E_1
  37. Dilate(A, If[0 < k ≤ 1, 1 - k, 0], D) → A'
  38. Dilate(A, If[0 < k ≤ 1, 1 - k, 0], M) → A'_1
  39. Dilate(A, If[0 < k ≤ 1, 1 - k, 0], E) → A'_2
  40. Dilate(A, If[0 < k ≤ 1, 1 - k, 0], O) → A'_3
  41. Angle(A, O, C_1) → α
  42. Angle(A, E, E_1) → β
  43. Angle(A, D, S) → δ
  44. Intersect(t, i, 1) → F_1
  45. Angle(C, I, F_1) → ζ
  46. Angle(C, G, W) → θ_1
  47. Dilate(C, If[0 < k ≤ 1, 1 - k, 0], G) → C'
  48. Dilate(C, If[0 < k ≤ 1, 1 - k, 0], F) → C'_1
  49. Dilate(C, If[0 < k ≤ 1, 1 - k, 0], I) → C'_2
  50. Dilate(C, If[0 < k ≤ 1, 1 - k, 0], K) → C'_3
  51. Intersect(i, j) → G_1
  52. Intersect(i, j) → H_1
  53. Line(C, H) → k_1
  54. Line(O, E) → l_1
  55. Intersect(e, ee) → J_1, , ,
  56. Intersect(e, l_1) → K_1, L_1
  57. Segment(H, K_1) → l
  58. Segment(C, L_1) → m
  59. Min(m, l) → n
  60. Circle(H, (0.3 * n)) → f_1
  61. Segment(C, F) → g_1
  62. Intersect(f_1, g) → I_1, M_1
  63. Intersect(f_1, l) → N_1,
  64. Line(F, C) → i_1
  65. Intersect(e, i_1) → O_1, P_1
  66. Angle(N_1, H, I_1) → ε
  67. Rotate(P_1, ε, C) → Q_1
  68. Angle(P_1, C, Q_1) → γ
  69. Dilate(F, If[0 < k ≤ 1, 1 - k, 0], C) → F'
  70. Segment(F, F') → h_1
  71. Dilate(C, If[1 < k ≤ 2, 2 - k, 0], P_1) → C'_4
  72. Segment(C, C'_4) → j_1
  73. Dilate(O, If[2 < k ≤ 3, 3 - k, 0], E) → O'
  74. Segment(O, O') → m_1
  75. Dilate(E, If[3 < k ≤ 4, 4 - k, 0], K_1) → E'
  76. Segment(E, E') → n_1
  77. Dilate(K_1, If[4 < k ≤ 5, 5 - k, 0], L_1) → R_1
  78. Segment(K_1, R_1) → p_1
  79. Dilate(H, If[5 < k ≤ 6, 6 - k, 0], K_1) → H'
  80. Segment(H, H') → q_1
  81. Line(O, K_1) → s_1
  82. Angle(P_1, C, L_1) → η
  83. Rotate(I_1, If[6 < k ≤ 7, ((-(k - 6)) * ε), (-ε)], H) → S_1
  84. CircleSector(H, S_1, I_1) → r_1
  85. Dilate(C, If[7 < k ≤ 8, 8 - k, 0], L_1) → C'_5
  86. Segment(C, C'_5) → t_1
  87. Dilate(H, If[8 < k ≤ 9, 9 - k, 0], C) → H'_1
  88. Vector(H, H'_1) → w
  89. Translate(r_1, w) → c_2
  90. Line(H, j_1) → a_1
  91. Line(H, t_1) → b_1
  92. Point(b_1) → T_1
  93. Angle(I_1, H, T_1) → θ
  94. Rotate(c_2, If[9 < k ≤ 10, ((k - 9) * θ), θ], C) → d_2
  95. Dilate(C, If[10 < k ≤ 11, 11 - k, 0], Q_1) → C'_6
  96. Segment(C, C'_6) → f_2
  97. Dilate(H, If[11 < k ≤ 12, 12 - k, 0], C) → H'_2
  98. Vector(H, H'_2) → u_1
  99. Translate(r_1, u_1) → e_2
  100. Rotate(e_2, If[12 < k ≤ 13, ((k - 12) * (θ - ε)), θ - ε], C) → g_2
  101. Segment(A, B) → h_2