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Dynamic Demonstration of Angle Bisectors in a Triangle with a 120-Degree Angle

Dynamic Demonstration of Angle Bisectors in a Triangle with a 120-Degree Angle
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2017-06-05 21:04
TriangleAngleGeometric Construction
This example dynamically illustrates the intersection points and geometric properties of the three angle bisectors in any triangle that contains a 120-degree angle. It involves core mathematical concepts such as triangles, angle bisectors, geometric transformations, and symmetry. The construction utilizes GeoGebra commands and objects like Rotate, Ray, Point, Line, Segment, Angle, Intersect, Min, Mirror, Circle, CircleSector, Polygon, OrthogonalLine, and Dilate. It is suitable for high school students and geometry enthusiasts to explore and understand the geometric properties of special triangles.

GeoGebra Commands & Parameters

  1. Rotate(B, π / 3, A) → B'
  2. Ray(A, B') → h
  3. Point(h) → C
  4. Line(A, C) → e
  5. Rotate(B', π / 3, A) → B''
  6. Ray(A, B'') → g
  7. Point(g) → U
  8. Ray(A, B) → j
  9. Point(j) → D
  10. Segment(A, D) → s
  11. Angle(A, U, D) → ζ
  12. Rotate(A, ζ / 2, U) → A'_1
  13. Ray(U, A'_1) → g_1
  14. Intersect(g_1, e) → K
  15. Segment(A, K) → i_1
  16. Intersect(g_1, s) → I
  17. Segment(A, I) → h_1
  18. Angle(U, D, A) → ε
  19. Rotate(U, ε / 2, D) → U'
  20. Ray(D, U') → f_1
  21. Intersect(f_1, g) → G
  22. Segment(A, G) → b
  23. Min({h_1, i_1, b}) → o
  24. Line(A, B) → f
  25. Angle(B, A, B') → α
  26. Angle(B', A, B'') → β
  27. Mirror(B, A) → B'_1
  28. Segment(C, D) → i
  29. Ray(A, B'_1) → k
  30. Segment(A, D) → l
  31. Segment(A, C) → m
  32. Circle(A, (0.55 * o)) → c
  33. Intersect(c, k, 1) → E
  34. Intersect(c, g) → F,
  35. CircleSector(A, F, E) → d
  36. Rotate(d, If[0 < a ≤ 1, ((-a) * π / 3), ((-π)) / 3], A) → d'
  37. Polygon(A, U, D) → poly1, a_1, c_1, d_1
  38. Rotate(d', If[2 < a ≤ 3, ((-(a - 2)) * π / 3), ((-π)) / 3], A) → d''
  39. Line(A, C) → n
  40. OrthogonalLine(D, n) → p
  41. Intersect(n, p) → J
  42. Line(A, D) → q
  43. OrthogonalLine(C, q) → r
  44. Intersect(r, q) → H
  45. Line(A, C) → n_1
  46. Circle(C, 1) → e_1
  47. Intersect(e_1, n_1, 2) → N
  48. Circle(A, 1) → p_1
  49. Intersect(p_1, n_1) → O, P
  50. Ray(C, N) → q_1
  51. Ray(A, O) → r_1
  52. Ray(A, D) → s_1
  53. Circle(D, 1) → t_1
  54. Intersect(t_1, s_1) → Q, R
  55. Ray(D, R) → b_1
  56. Ray(D, U) → j_2
  57. Ray(A, C) → t
  58. Angle(U, D, U') → η
  59. Angle(A, U, A'_1) → θ
  60. Intersect(e, j_2) → L
  61. Segment(D, U) → j_1
  62. Segment(D, I) → k_1
  63. Segment(D, L) → f_2
  64. Segment(D, K) → g_2
  65. Segment(U, L) → h_2
  66. Segment(U, K) → i_2
  67. Segment(U, G) → k_2
  68. Line(A, D) → l_2
  69. Min({s, f_2, g_2, h_2, i_2, k_2}) → u
  70. Circle(D, (0.4 * u)) → c_2
  71. Circle(U, (0.4 * u)) → d_2
  72. Intersect(d_2, g) → S, T
  73. Intersect(d_2, g_1) → V,
  74. Intersect(c_2, j_1) → W,
  75. Intersect(c_2, f_1) → Z,
  76. CircleSector(D, W, Z) → e_2
  77. CircleSector(U, S, V) → p_2
  78. Dilate(D, If[3 < a ≤ 4, 4 - a, 0], G) → D'
  79. Segment(D, D') → m_2
  80. Dilate(U, If[4 < a ≤ 5, 5 - a, 0], I) → U'_1
  81. Segment(U, U'_1) → n_2
  82. Rotate(e_2, If[5 < a ≤ 6, ((a - 5) * η), η], D) → q_2
  83. Rotate(p_2, If[6 < a ≤ 7, ((a - 6) * θ), θ], U) → r_2
  84. Ray(G, L) → s_2
  85. Ray(I, L) → t_2
  86. Intersect(n_2, s_2) → A_1
  87. Line(U, I) → a_2
  88. Intersect(e, a_2) → B_1
  89. Intersect(t_2, m_2) → C_1
  90. Segment(L, A_1) → b_2
  91. Segment(L, B_1) → f_3
  92. Segment(L, C_1) → g_3
  93. Min({b_2, f_3, g_3}) → v
  94. Circle(L, (0.4 * v)) → c_3
  95. Intersect(c_3, s_2) → D_1, E_1
  96. Intersect(c_3, t_2) → F_1, G_1
  97. CircleSector(L, D_1, F_1) → d_3
  98. Dilate(G, If[7 < a ≤ 8, 8 - a, 0], L) → G'
  99. Dilate(I, If[7 < a ≤ 8, 8 - a, 0], L) → I'
  100. Segment(G, G') → l_1
  101. Segment(I, I') → m_1
  102. Rotate(d_3, If[9 < a ≤ 10, ((a - 9) * π / 2), π / 2], L) → e_3