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Dynamic Demo of Right Triangle's Incircle and Excircle

Dynamic Demo of Right Triangle's Incircle and Excircle
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2019-07-31 03:39
TriangleCircleGeometric Construction
This example dynamically illustrates the geometric relationships between the incircle and excircles of a right triangle. Key concepts include right triangles, incircles, excircles, tangent lines, rotations, and similar triangles. Construction involves GeoGebra commands like Intersect, Semicircle, Polygon, Rotate, Dilate, etc. Suitable for high school geometry education to understand properties of special quadrilaterals and circles.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Semicircle(A, B) → c
  3. Point(c) → C
  4. Polygon(A, C, B) → t1, b, a_1, c_1
  5. Angle(B, A, C) → α
  6. Rotate(B, α / 2, A) → B'
  7. Angle(B, A, B') → β
  8. Ray(A, B') → f
  9. Rotate(A, π / 4, C) → A'
  10. Ray(C, A') → g
  11. Ray(A, C) → h
  12. Ray(A, B) → i
  13. Segment(B, C) → j
  14. Intersect(g, f) → D
  15. Ray(B, B + (1, 0, 0)) → k
  16. Point(k) → E
  17. Circle(C, 3) → d
  18. Intersect(d, h, 2) → F
  19. Angle(E, B, C, xOyPlane) → γ
  20. Rotate(E, γ / 2, B) → E'
  21. Ray(B, E') → l
  22. Rotate(B, π / 4, C) → B'_1
  23. Ray(C, B'_1) → m
  24. Intersect(l, m) → G
  25. OrthogonalLine(D, h) → n
  26. OrthogonalLine(D, k, xOyPlane) → p
  27. Intersect(n, h) → H
  28. Intersect(p, i) → I
  29. OrthogonalLine(G, h, xOyPlane) → q
  30. OrthogonalLine(G, k, xOyPlane) → r
  31. Intersect(q, h) → J
  32. Intersect(r, i) → K
  33. Circle(G, J, xOyPlane) → e
  34. Circle(D, H) → s
  35. Ray(B, D) → t
  36. Angle(A, C, B) → δ
  37. Ray(B, D) → f_1
  38. Ray(A, D) → g_1
  39. Min({Distance[A, D], Distance[B, D], Distance[B, G], Distance[C, G]}) → o
  40. Circle(A, (0.35 * o)) → d_1
  41. Intersect(d_1, i) → L,
  42. Circle(B, (0.35 * o)) → e_1
  43. Circle(C, (0.35 * o)) → h_1
  44. Intersect(h_1, h, 2) → M
  45. Intersect(k, e_1) → N,
  46. Intersect(e_1, j) → O,
  47. Dilate(A, If[0 < u ≤ 1, 1 - u, 0], D) → A'_1
  48. Segment(A, A'_1) → i_1
  49. Dilate(B, If[0 < u ≤ 1, 1 - u, 0], D) → B'_2
  50. Segment(B, B'_2) → j_1
  51. Rotate(O, If[0 < u ≤ 1, (u * (π / 2 - α) / 2), (π / 2 - α) / 2], B) → O'
  52. Rotate(L, If[0 < u ≤ 1, (u * β), β], A) → L'
  53. CircleSector(B, O, O') → k_1
  54. CircleSector(A, L, L') → p_1
  55. Dilate(C, If[1 < u ≤ 2, 2 - u, 0], G) → C'
  56. Dilate(B, If[1 < u ≤ 2, 2 - u, 0], G) → B'_3
  57. Segment(B, B'_3) → l_1
  58. Segment(C, C') → m_1
  59. Intersect(h_1, j) → P,
  60. Rotate(P, If[1 < u ≤ 2, ((u - 1) * π / 4), π / 4], C) → P'
  61. Rotate(N, If[1 < u ≤ 2, ((u - 1) * γ / 2), γ / 2], B) → N'
  62. CircleSector(B, N, N', xOyPlane) → q_1
  63. CircleSector(C, P, P') → r_1
  64. Rotate(p_1, If[2.5 < u ≤ 3.5, ((u - 2.5) * β), β], A) → s_1
  65. Rotate(k_1, If[2.5 < u ≤ 3.5, ((u - 2.5) * (π / 2 - α) / 2), (π / 2 - α) / 2], B) → t_1
  66. Rotate(q_1, If[3.5 < u ≤ 4.5, ((u - 3.5) * γ / 2), γ / 2], B) → c_2
  67. Rotate(r_1, If[3.5 < u ≤ 4.5, ((u - 3.5) * π / 4), π / 4], C) → d_2
  68. Dilate(D, If[5 < u ≤ 6, 6 - u, 0], H) → D'
  69. Circle(D, D') → e_2
  70. OrthogonalLine(D, j) → n_1
  71. Intersect(j, n_1) → Q
  72. Dilate(G, If[6 < u ≤ 7, 7 - u, 0], J) → G'
  73. Circle(G, G', xOyPlane) → f_2
  74. OrthogonalLine(G, j, xOyPlane) → b_1
  75. Intersect(b_1, j) → R
  76. Dilate(H, If[12 < u ≤ 13, 13 - u, 0], R) → H'
  77. Dilate(R, If[13 < u ≤ 14, 14 - u, 0], K) → R'
  78. Ray(C, J) → i_2
  79. Segment(A, C) → j_2
  80. Dilate(D, If[16 < u ≤ 17, 17 - u, 0], R) → D'_1
  81. Ray(K, R) → m_2
  82. Ray(J, Q) → n_2
  83. Intersect(m_2, n_2) → U
  84. Dilate(K, If[12 < u ≤ 13, 13 - u, 0], R) → K'
  85. Dilate(J, If[12 < u ≤ 13, 13 - u, 0], Q) → J'
  86. Segment(J, J') → p_2
  87. Segment(K, K') → q_2
  88. Dilate(R, If[13 < u ≤ 14, 14 - u, 0], U) → R'_1
  89. Dilate(Q, If[13 < u ≤ 14, 14 - u, 0], U) → Q'
  90. Segment(Q, Q') → g_2
  91. Segment(R, R'_1) → k_2
  92. Dilate(J, If[14 < u ≤ 15, 15 - u, 0], K) → J'_1
  93. Segment(J, J'_1) → l_2
  94. Polygon(J, U, K) → t2, k_3, j_3, u_1
  95. Dilate(C, If[16 < u ≤ 17, 17 - u, 0], J) → C'_1
  96. Vector(C, C'_1) → v
  97. Translate(d_2, v) → h_2
  98. Line(C, p_2) → r_2
  99. Point(r_2) → S
  100. Angle(S, C, G, xOyPlane) → ε
  101. Rotate(h_2, If[16 < u ≤ 17, ((-(u - 16)) * ε), (-ε)], C'_1) → s_2
  102. Dilate(J, If[18 < u ≤ 19, 19 - u, 0], K) → J'_2
  103. Vector(J, J'_2) → w
  104. Translate(s_2, w) → t_2
  105. Line(K, p_2) → a_2
  106. Point(a_2) → T
  107. Rotate(t_2, If[18 < u ≤ 19, (((-(u - 18)) * 3) * π / 4), (-3 * π / 4)], J'_2) → c_3