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Dynamic Illustration of External/Internal Tangents Theorem

Dynamic Illustration of External/Internal Tangents Theorem
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2016-12-18 04:23
CircleVectorTransformationGeometric Construction
This example dynamically illustrates a less-known theorem regarding segments formed by external and internal tangents of two circles, serving as a follow-up to previous explorations. It involves core mathematical concepts such as circles, tangent lines, similarity transformations, vector operations, and more. The construction utilizes commands like Circle, Segment, PointIn, Distance, Tangent, Intersect, Dilate, Angle, Midpoint, Rotate, Vector, Translate, Line, and Point in GeoGebra. Suitable for high school and above students to explore geometric transformations and theorem validation.

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[4 < k ≤ 5, 5 - k, 0], M) → A'_1
  39. Dilate(A, If[9 < k ≤ 10, 10 - 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, J, Q) → γ
  44. Angle(A, D, S) → δ
  45. Angle(C, K, A_1) → ε
  46. Intersect(t, i, 1) → F_1
  47. Angle(C, I, F_1) → ζ
  48. Angle(C, F, T) → η
  49. Angle(C, G, W) → θ
  50. Dilate(C, If[0 < k ≤ 1, 1 - k, 0], G) → C'
  51. Dilate(C, If[9 < k ≤ 10, 10 - k, 0], F) → C'_1
  52. Dilate(C, If[4 < k ≤ 5, 5 - k, 0], I) → C'_2
  53. Dilate(C, If[0 < k ≤ 1, 1 - k, 0], K) → C'_3
  54. Segment(A, A') → l
  55. Segment(A, A'_1) → m
  56. Segment(A, A'_2) → n
  57. Segment(A, A'_3) → f_1
  58. Segment(C, C') → g_1
  59. Segment(C, C'_1) → h_1
  60. Segment(C, C'_2) → i_1
  61. Segment(C, C'_3) → j_1
  62. Intersect(i, j) → G_1
  63. Intersect(i, j) → H_1
  64. Intersect(h, i) → I_1
  65. Intersect(g, j) → J_1
  66. Intersect(g, i) → K_1
  67. Intersect(h, j) → L_1
  68. Segment(I_1, M) → k_1
  69. Segment(M, I) → l_1
  70. Segment(I, K_1) → m_1
  71. Segment(J_1, E) → n_1
  72. Segment(E, F) → p_1
  73. Segment(F, L_1) → q_1
  74. Dilate(K_1, If[1 < k ≤ 2, 2 - k, 0], M) → M_1
  75. Dilate(M, If[2 < k ≤ 3, 3 - k, 0], I) → M'
  76. Dilate(I, If[3 < k ≤ 4, 4 - k, 0], I_1) → I'
  77. Segment(K_1, M_1) → r_1
  78. Segment(M, M') → s_1
  79. Segment(I, I') → t_1
  80. Midpoint(M, I) → N_1
  81. Rotate(K_1, If[5 < k ≤ 6, ((k - 5) * π), π], N_1) → O_1
  82. Rotate(M_1, If[5 < k ≤ 6, ((k - 5) * π), π], N_1) → P_1
  83. Segment(O_1, P_1) → a_1
  84. Dilate(J_1, If[6 < k ≤ 7, 7 - k, 0], F) → Q_1
  85. Dilate(F, If[7 < k ≤ 8, 8 - k, 0], E) → F'
  86. Dilate(E, If[8 < k ≤ 9, 9 - k, 0], L_1) → E'
  87. Segment(J_1, Q_1) → b_1
  88. Segment(F, F') → f_2
  89. Segment(E, E') → g_2
  90. Midpoint(Q_1, E) → R_1
  91. Rotate(J_1, If[10 < k ≤ 11, ((k - 10) * π), π], R_1) → S_1
  92. Rotate(Q_1, If[10 < k ≤ 11, ((k - 10) * π), π], R_1) → T_1
  93. Segment(S_1, T_1) → h_2
  94. Dilate(L_1, If[11 < k ≤ 12, 12 - k, 0], I_1) → U_1
  95. Vector(L_1, U_1) → u
  96. Translate(S_1, u) → V_1
  97. Translate(T_1, u) → W_1
  98. Segment(V_1, W_1) → i_2
  99. Line(I_1, f_2) → j_2
  100. Point(j_2) → Z_1
  101. Angle(Z_1, I_1, M') → ι
  102. Rotate(V_1, If[12 < k ≤ 13, ((k - 12) * ι), ι], I_1) → A_2
  103. Rotate(W_1, If[12 < k ≤ 13, ((k - 12) * ι), ι], I_1) → B_2
  104. Segment(A_2, B_2) → k_2
  105. Midpoint(E, M_1) → C_2
  106. Rotate(S_1, If[13 < k ≤ 14, ((-(k - 13)) * π), (-π)], C_2) → D_2
  107. Rotate(T_1, If[13 < k ≤ 14, ((-(k - 13)) * π), (-π)], C_2) → E_2
  108. Segment(D_2, E_2) → l_2
  109. Line(M_1, f_2) → m_2
  110. Point(m_2) → F_2
  111. Angle(K_1, M_1, F_2) → κ
  112. Rotate(D_2, If[14 < k ≤ 15, ((-(k - 14)) * κ), (-κ)], M_1) → G_2
  113. Rotate(E_2, If[14 < k ≤ 15, ((-(k - 14)) * κ), (-κ)], M_1) → H_2
  114. Segment(G_2, H_2) → n_2

AI Drawing Prompt Reference

Draw two intersecting circles and find their common tangents and internal tangents. Create small circles at the intersection points of these tangents, and use these circles and points to construct a series of new shapes involving rotations, dilations, and translations. Measure and display key angles to illustrate similarity and vector relationships, resulting in a dynamic illustration of a geometric theorem validation.
The prompt above was compiled from the original drawing commands and is for reference only.