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Tangent–Circle Configuration via Geometric Transformations

Tangent–Circle Configuration via Geometric Transformations
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2023-03-13 04:38
CircleFunctionTrigonometryTransformationGeometric Construction
In collections geometriadinamica.es
This example demonstrates how to build a tangent–circle configuration by combining circle construction with tangents, intersections, and a sequence of geometric transformations. Visually, it highlights key points such as the tangency/contact points and the intersection structure created under given constraints. The central math ideas include circle geometry, tangent lines, line–circle (or line–line) intersection, and the broader notion of relating geometric proportions/angles to functional relationships (consistent with the provided topic tags such as exponent/cosine). Construction uses GeoGebra commands Ray to set directions, Dilate to scale key elements, Rotate to reorient the configuration, and OrthogonalLine to enforce perpendicular constraints. It employs Polygon and Segment to organize the framework, Circle to define the main circle, Tangent to generate tangents, and Intersect to compute critical intersection points. The construction also relies on multiple parameterized points/variables (e.g., A–K, along with parameters like k, a, b, c, and radio) to drive the geometry and measurements. Suitable for late middle school through early university, especially for lessons on geometric construction, tangent properties, and transformation-based reasoning.

GeoGebra Commands & Parameters

  1. Ray(A, C) → a
  2. Ray(A, B) → b
  3. Ray(A, D) → c
  4. Dilate(B, k, A) → E
  5. Dilate(C, k, A) → F
  6. Dilate(D, k, A) → G
  7. Polygon(A, E, G) → P, g, a_1, e
  8. Polygon(A, F, E) → Q, e_1, a_2, f
  9. Dilate(H, k, A) → I
  10. Polygon(A, G, I) → R, i, a_3, g_1
  11. Polygon(A, F, I) → S, i_1, a_4, f_1
  12. Circle(J, radio) → d
  13. Dilate(J, k, A) → K
  14. Circle(K, (radio * k)) → h
  15. Tangent(A, h) → j, l
  16. Intersect(h, l) → L
  17. Intersect(h, j) → M
  18. Segment(L, A) → m
  19. Segment(A, M) → n
  20. Point(h) → N
  21. Rotate(N, α, K) → O
  22. Rotate(O, α, K) → T
  23. Rotate(T, α, K) → U
  24. Rotate(U, α, K) → V
  25. Rotate(V, α, K) → W
  26. Rotate(W, α, K) → Y
  27. Rotate(Y, α, K) → Z
  28. Rotate(Z, α, K) → A_1
  29. Rotate(A_1, α, K) → B_1
  30. Rotate(B_1, α, K) → C_1
  31. Rotate(C_1, α, K) → D_1
  32. Rotate(D_1, α, K) → E_1
  33. Rotate(E_1, α, K) → F_1
  34. Rotate(F_1, α, K) → G_1
  35. Rotate(G_1, α, K) → H_1
  36. Rotate(H_1, α, K) → I_1
  37. Rotate(I_1, α, K) → J_1
  38. Rotate(J_1, α, K) → K_1
  39. Rotate(K_1, α, K) → L_1
  40. Rotate(L_1, α, K) → M_1
  41. Rotate(M_1, α, K) → N_1
  42. Rotate(N_1, α, K) → O_1
  43. Rotate(O_1, α, K) → P_1
  44. Rotate(P_1, α, K) → Q_1
  45. Rotate(Q_1, α, K) → R_1
  46. Rotate(R_1, α, K) → S_1
  47. Rotate(S_1, α, K) → T_1
  48. Rotate(T_1, α, K) → U_1
  49. Rotate(U_1, α, K) → V_1
  50. Rotate(V_1, α, K) → W_1
  51. Rotate(W_1, α, K) → Y_1
  52. Rotate(Y_1, α, K) → Z_1
  53. Segment(K, N) → p
  54. Segment(K, Z_1) → q
  55. Segment(K, Y_1) → r
  56. Segment(K, W_1) → s
  57. Segment(K, V_1) → t
  58. Segment(K, U_1) → b_1
  59. Segment(K, T_1) → c_1
  60. Segment(K, S_1) → d_1
  61. Segment(K, R_1) → h_1
  62. Segment(K, Q_1) → j_1
  63. Segment(K, P_1) → k_1
  64. Segment(O_1, K) → l_1
  65. Segment(N_1, K) → m_1
  66. Segment(M_1, K) → n_1
  67. Segment(L_1, K) → p_1
  68. Segment(K, K_1) → q_1
  69. Segment(K, J_1) → r_1
  70. Segment(K, I_1) → s_1
  71. Segment(K, H_1) → t_1
  72. Segment(K, G_1) → b_2
  73. Segment(K, F_1) → c_2
  74. Segment(K, E_1) → d_2
  75. Segment(K, D_1) → e_2
  76. Segment(K, C_1) → f_2
  77. Segment(K, B_1) → g_2
  78. Segment(K, A_1) → h_2
  79. Segment(K, Z) → i_2
  80. Segment(K, Y) → j_2
  81. Segment(K, W) → k_2
  82. Segment(K, V) → l_2
  83. Segment(K, U) → m_2
  84. Segment(K, T) → n_2
  85. Segment(K, O) → p_2
  86. OrthogonalLine(A_2, yAxis) → q_2
  87. Point(q_2) → B_2
  88. Dilate(B_2, k, A_2) → C_2
  89. OrthogonalLine(B_2, q_2) → r_2
  90. Point(r_2) → D_2
  91. Circle(B_2, D_2) → s_2
  92. Ray(A_2, D_2) → t_2
  93. Intersect(s_2, r_2, 2) → E_2
  94. Ray(A_2, E_2) → b_3
  95. OrthogonalLine(C_2, q_2) → c_3
  96. Intersect(t_2, c_3) → F_2
  97. Intersect(b_3, c_3) → G_2
  98. Segment(F_2, G_2) → d_3
  99. Segment(A_2, F_2) → e_3
  100. Segment(A_2, G_2) → f_3
  101. Segment(A_2, C_2) → g_3

AI Drawing Prompt Reference

Start from a common vertex, draw three rays in different directions)Skip, and scale points along each ray by the same factor to form two adjacent triangles with the vertex. Then construct a circle centered at another point, and dilate its center by the same factor to create a concentric circle. From the vertex, draw two tangents to the inner circle, mark the tangent points, and connect them to the vertex with segments. Finally, take a point on the inner circle and rotate it repeatedly around the center by a fixed angle to generate a symmetric ring of points. The final figure combines circles, tangents, triangles, and a rotated point pattern.
The prompt above was compiled from the original drawing commands and is for reference only.