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Square, Tangent Circles, and External Tangent

Square, Tangent Circles, and External Tangent
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2016-12-09 10:33
CirclePolygonVectorCoordinate Geometry
This example demonstrates the geometric relationship between a square, two tangent circles, and their common external tangent. It involves core math concepts like circles, polygons, angles, vectors, functions, coordinate geometry, and geometric constructions. The construction utilizes GeoGebra commands and objects such as Circle, Segment, Intersect, Angle, Dilate, Rotate, Line, Midpoint, Polygon, Vector, and Translate. It is suitable for high school students learning geometric constructions and properties, particularly in the context of coordinate geometry and geometric constructions.

GeoGebra Commands & Parameters

  1. Circle(A, D) → c
  2. Circle(B, F_2) → d
  3. Segment(H_1, G_1) → f
  4. Circle(G_1, f) → e
  5. Circle(H_1, f) → g
  6. Intersect(c, g, 1) → M
  7. Intersect(d, e, 2) → H
  8. Angle(M, H, G_1) → α
  9. Angle(H, G_1, E_2) → β
  10. Angle(G_1, H_1, M) → γ
  11. Angle(H_1, M, H) → δ
  12. Segment(M, H) → h
  13. Segment(H, G_1) → i
  14. Segment(M, H_1) → j
  15. Dilate(A, If[0 < b ≤ 1, 1 - b, 0], D) → A'
  16. Segment(A, A') → l
  17. Rotate(A, If[1 < b ≤ 2, ((b - 1) * π / 2), π / 2], A) → A'_1
  18. Rotate(A', If[1 < b ≤ 2, ((b - 1) * π / 2), π / 2], A) → A''
  19. Segment(A'_1, A'') → l'
  20. Rotate(A'_1, If[2 < b ≤ 3, ((b - 2) * π), π], C) → K
  21. Rotate(A'', If[2 < b ≤ 3, ((b - 2) * π), π], C) → A'''
  22. Segment(K, A''') → l''
  23. Line(B, F_2) → m
  24. Circle(F_2, l / 5) → p
  25. Intersect(p, m) → L, M_1
  26. Circle(L, l / 5) → q
  27. Intersect(q, m) → N, O
  28. Circle(N, l / 5) → r
  29. Intersect(r, m) → P, Q
  30. Midpoint(P, B) → R
  31. Midpoint(M, H) → S
  32. Dilate(M, If[4 < b ≤ 5, 5 - b, 0], S) → I'
  33. Segment(M, I') → n
  34. Dilate(S, If[5 < b ≤ 6, 6 - b, 0], H) → S'
  35. Segment(S, S') → s
  36. Rotate(M, If[7 < b ≤ 8, ((b - 7) * π), π], S) → I'_1
  37. Rotate(I', If[7 < b ≤ 8, ((b - 7) * π), π], S) → I''
  38. Rotate(M, If[6 < b ≤ 7, ((b - 6) * π), π], S) → I'_2
  39. Rotate(I', If[6 < b ≤ 7, ((b - 6) * π), π], S) → I''_1
  40. Segment(I'_2, I''_1) → n'
  41. Rotate(M, If[7 < b ≤ 8, ((b - 7) * π / 2), π / 2], H) → G
  42. Rotate(I', If[7 < b ≤ 8, ((b - 7) * π / 2), π / 2], H) → I''_2
  43. Segment(G, I''_2) → n'_1
  44. Rotate(I'_2, If[7 < b ≤ 8, ((b - 7) * π / 2), π / 2], H) → T
  45. Rotate(I''_1, If[7 < b ≤ 8, ((b - 7) * π / 2), π / 2], H) → U
  46. Segment(T, U) → n''
  47. Rotate(S, If[7 < b ≤ 8, ((b - 7) * π / 2), π / 2], H) → S'_1
  48. Angle(M, H, G_1) → ε
  49. Rotate(T, If[8 < b ≤ 9, ((b - 8) * π / 2), π / 2], G_1) → F
  50. Rotate(U, If[8 < b ≤ 9, ((b - 8) * π / 2), π / 2], G_1) → U'
  51. Segment(F, U') → n'''
  52. Rotate(G, If[8 < b ≤ 9, ((b - 8) * π / 2), π / 2], G_1) → V
  53. Rotate(I''_2, If[8 < b ≤ 9, ((b - 8) * π / 2), π / 2], G_1) → W
  54. Segment(V, W) → t
  55. Rotate(S'_1, If[8 < b ≤ 9, ((b - 8) * π / 2), π / 2], G_1) → Z
  56. Angle(H, G_1, H_1) → ζ
  57. Rotate(F, If[9 < b ≤ 10, ((b - 9) * π / 2), π / 2], H_1) → T''
  58. Rotate(U', If[9 < b ≤ 10, ((b - 9) * π / 2), π / 2], H_1) → U''
  59. Segment(T'', U'') → f_1
  60. Rotate(Z, If[9 < b ≤ 10, ((b - 9) * π / 2), π / 2], H_1) → Z'
  61. Rotate(V, If[9 < b ≤ 10, ((b - 9) * π / 2), π / 2], H_1) → V'
  62. Rotate(W, If[9 < b ≤ 10, ((b - 9) * π / 2), π / 2], H_1) → W'
  63. Segment(V', W') → t'
  64. Polygon(H_1, G_1, H, M) → poly1, h_1, g_1, j_1, i_1
  65. Angle(G_1, H_1, M) → η
  66. Dilate(H_1, If[10 < b ≤ 11, 11 - b, 0], D) → H'
  67. Dilate(G_1, If[10 < b ≤ 11, 11 - b, 0], F_2) → E
  68. Segment(G_1, E) → k_1
  69. Segment(H_1, H') → l_1
  70. Angle(H_1, D, A) → θ
  71. Dilate(F_2, If[11 < b ≤ 12, 12 - b, 0], N) → F'
  72. Segment(F_2, F') → m_1
  73. Dilate(N, If[12 < b ≤ 13, 13 - b, 0], R) → N'
  74. Segment(N, N') → n_1
  75. Dilate(R, If[13 < b ≤ 14, 14 - b, 0], B) → R'
  76. Segment(R, R') → p_1
  77. Dilate(H, If[14 < b ≤ 15, 15 - b, 0], N) → J'
  78. Vector(H, J') → u
  79. Translate(T, u) → T'_1
  80. Translate(U, u) → U'_1
  81. Segment(T'_1, U'_1) → n'''_1
  82. Translate(S'_1, u) → A_1
  83. Translate(G, u) → B_1
  84. Translate(I''_2, u) → C_1
  85. Segment(B_1, C_1) → q_1
  86. Dilate(H, If[15 < b ≤ 16, 16 - b, 0], R) → J'_1
  87. Vector(H, J'_1) → v
  88. Translate(S'_1, v) → D_1
  89. Translate(T, v) → T'_2
  90. Translate(U, v) → U'_2
  91. Segment(T'_2, U'_2) → n'''_2
  92. Translate(G, v) → E_1
  93. Translate(I''_2, v) → F_1
  94. Segment(E_1, F_1) → r_1
  95. Dilate(H, If[16 < b ≤ 17, 17 - b, 0], B) → J'_2
  96. Vector(H, J'_2) → w
  97. Translate(T, w) → T'_3
  98. Translate(U, w) → U'_3
  99. Segment(T'_3, U'_3) → n'''_3
  100. Angle(B, F_2, G) → ι