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Dynamic Incircle and Excircle Relationship

Dynamic Incircle and Excircle Relationship
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2017-04-22 23:33
TriangleCircleTrigonometryGeometric Construction
This example dynamically illustrates the relationship between a triangle's incircle and one of its excircles. It involves core mathematical concepts such as circles, triangles, angles, and geometric transformations. The construction utilizes commands like Rotate, Ray, Point, Segment, Min, Angle, Circle, Intersect, CircleSector, OrthogonalLine, Midpoint, Polygon, Dilate, CircleArc, Vector, Translate, and Line. It is suitable for high school and above students to understand the geometric properties and dynamic relationships between triangles and circles.

GeoGebra Commands & Parameters

  1. Rotate(B, (a * π / 180), A) → B'
  2. Ray(A, B') → g
  3. Point(g) → C
  4. Segment(A, C) → j
  5. Ray(A, B) → f
  6. Point(f) → D
  7. Segment(C, D) → h
  8. Segment(A, D) → i
  9. Min({i, j, h}) → b
  10. Angle(B, A, B') → α
  11. Circle(D, (0.2 * b)) → d
  12. Circle(C, 4) → c
  13. Intersect(c, g, 2) → E
  14. Intersect(d, f, 2) → F
  15. Ray(D, F) → k
  16. Ray(C, E) → l
  17. Circle(C, (0.2 * b)) → e
  18. Circle(D, (0.15 * b)) → p
  19. Angle(F, D, C) → β
  20. Angle(D, C, E) → γ
  21. Rotate(F, β / 2, D) → F'
  22. Angle(F, D, F') → δ
  23. Rotate(D, γ / 2, C) → D'
  24. Angle(D, C, D') → ε
  25. Ray(C, D') → m
  26. Ray(D, F') → n
  27. Intersect(e, m, 1) → G
  28. Intersect(e, h, 1) → H
  29. Intersect(p, h, 1) → I
  30. CircleSector(C, H, G) → q
  31. CircleSector(D, F, F') → r
  32. Intersect(m, n) → J
  33. OrthogonalLine(J, k) → s
  34. OrthogonalLine(J, l) → t
  35. Intersect(s, k) → K
  36. Intersect(t, l) → L
  37. Circle(J, K) → c_1
  38. Segment(L, K) → f_1
  39. Midpoint(f_1) → M
  40. OrthogonalLine(M, f_1) → g_1
  41. Intersect(c_1, g_1, 2) → N
  42. Segment(N, C) → h_1
  43. Segment(N, D) → i_1
  44. Polygon(A, D, C) → poly1, c_2, a_1, d_2
  45. Angle(J, K, D) → ζ
  46. Angle(C, L, J) → η
  47. Dilate(A, If[0 < o ≤ 1, 1 - o, 0], D) → A'
  48. Dilate(A, If[0 < o ≤ 1, 1 - o, 0], C) → A'_1
  49. Segment(A, A') → k_1
  50. Segment(A, A'_1) → l_1
  51. Dilate(C, If[1 < o ≤ 2, 2 - o, 0], J) → C'
  52. Dilate(D, If[1 < o ≤ 2, 2 - o, 0], J) → D'_1
  53. Segment(C, C') → m_1
  54. Segment(D, D'_1) → n_1
  55. Rotate(r, If[2.5 < o ≤ 3.5, ((o - 2.5) * δ), δ], D) → r'
  56. Rotate(q, If[3.5 < o ≤ 4.5, ((o - 3.5) * ε), ε], C) → q'
  57. Dilate(J, If[5 < o ≤ 6, 6 - o, 0], L) → J'
  58. Dilate(J, If[5 < o ≤ 6, 6 - o, 0], K) → J'_1
  59. Segment(J, J'_1) → p_1
  60. Segment(J, J') → q_1
  61. Angle(L, J, K) → θ
  62. Rotate(L, If[6 < o ≤ 7, ((o - 6) * θ), θ], J) → L'
  63. CircleArc(J, L, L') → e_1
  64. Rotate(K, If[7 < o ≤ 8, ((o - 7) * ((2 * π) - θ) / 2), ((2 * π) - θ) / 2], J) → K'
  65. CircleArc(J, K, K') → r_1
  66. Rotate(N, If[8 < o ≤ 9, ((o - 8) * ((2 * π) - θ) / 2), ((2 * π) - θ) / 2], J) → N'
  67. CircleArc(J, N, N') → s_1
  68. Dilate(N, If[11 < o ≤ 12, 12 - o, 0], D) → N'_1
  69. Dilate(N, If[11 < o ≤ 12, 12 - o, 0], C) → N'_2
  70. Min(c_2, d_2) → w
  71. Circle(A, (0.15 * w)) → e_2
  72. Segment(A, C) → m_2
  73. Intersect(e_2, m_2) → S,
  74. Segment(A, D) → n_2
  75. Intersect(e_2, n_2) → T,
  76. Min({h, k_1, l_1}) → b_1
  77. Intersect(d, h, 1) → U
  78. Intersect(e, h) → W,
  79. Angle(A, C, D) → λ
  80. Angle(C, D, A) → κ
  81. Rotate(A, λ / 2, C) → A'_2
  82. Angle(A, C, A'_2) → ι
  83. Ray(C, A'_2) → t_1
  84. Rotate(C, κ / 2, D) → C'_1
  85. Angle(C, D, C'_1) → μ_1
  86. Ray(D, C'_1) → f_2
  87. Intersect(d, f_2) → A_1,
  88. CircleSector(D, U, A_1) → r_2
  89. Intersect(t_1, f_2) → Z
  90. OrthogonalLine(Z, k_1) → s_2
  91. OrthogonalLine(Z, l_1) → t_2
  92. Intersect(s_2, k_1) → B_1
  93. Intersect(t_2, l_1) → C_1
  94. Intersect(k_1, t_1) → D_1
  95. Circle(A, (0.2 * b)) → q_2
  96. Angle(D, A, C) → μ
  97. Rotate(D, μ / 2, A) → D'_2
  98. Angle(D, A, D'_2) → ν
  99. Ray(A, D'_2) → a_2
  100. Intersect(q_2, a_2) → V,
  101. Intersect(q_2, k_1) → E_1,
  102. CircleSector(A, E_1, V) → p_2
  103. Dilate(D, If[1 < o ≤ 2, 2 - o, 0], Z) → D'_3
  104. Dilate(A, If[1 < o ≤ 2, 2 - o, 0], Z) → A'_3
  105. Segment(A, A'_3) → b_2
  106. Segment(D, D'_3) → f_3
  107. Rotate(r_2, If[2.5 < o ≤ 3.5, ((o - 2.5) * κ / 2), κ / 2], D) → c_3
  108. Rotate(p_2, If[3.5 < o ≤ 4.5, ((o - 3.5) * μ / 2), μ / 2], A) → d_3
  109. Dilate(Z, If[5 < o ≤ 6, 6 - o, 0], C_1) → Z'
  110. Dilate(Z, If[5 < o ≤ 6, 6 - o, 0], B_1) → Z'_1
  111. Segment(Z, Z') → g_3
  112. Segment(Z, Z'_1) → h_3
  113. Angle(Z, B_1, A) → ξ
  114. Angle(A, C_1, Z) → ο
  115. Circle(Z, Z') → e_3
  116. Rotate(C_1, If[6 < o ≤ 9, (((o - 6) * 2) * π / 3), (2 * π)], Z) → F_1
  117. Segment(C, D_1) → i_3
  118. Intersect(e, i_3) → G_1,
  119. Intersect(e, l_1) → H_1,
  120. Dilate(C, If[9.5 < o ≤ 10.5, 10.5 - o, 0], Z) → C'_2
  121. Dilate(J, If[9.5 < o ≤ 10.5, 10.5 - o, 0], Z) → J'_2
  122. Segment(J, J'_2) → j_1
  123. Segment(C, C'_2) → d_1
  124. CircleSector(C, H_1, G_1) → g_2
  125. Polygon(A, Z, C) → poly2, c_4, a_3, z_1
  126. Polygon(A, D, J) → poly3, j_2, a_4, d_4
  127. CircleArc(A, E_1, V) → h_2
  128. CircleArc(Z, Z', F_1) → k_3
  129. Dilate(C, If[11 < o ≤ 12, 12 - o, 0], J) → C'_3
  130. Vector(C, C'_3) → u_1
  131. Translate(g_2, u_1) → k_2
  132. Line(C, j_1) → i_2
  133. Point(i_2) → O
  134. Angle(O, C, H_1) → ρ
  135. Rotate(k_2, If[12 < o ≤ 13, ((-(o - 12)) * ρ), (-ρ)], J) → p_3

AI Drawing Prompt Reference

Draw a triangle and dynamically illustrate the relationship between its incircle and one of its excircles. Start by drawing a side of the triangle, then find the midpoint of this side and draw an incircle centered at this point. Next, draw an excircle centered at the midpoint of the side opposite the triangle's vertex, with the radius equal to the distance from the midpoint to the vertex. Connect the intersection points of the incircle and excircle to show the dynamic geometric relationship.
The prompt above was compiled from the original drawing commands and is for reference only.