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Dynamic Illustration of Geometric Relations in a Triangle with Isosceles Triangl

Dynamic Illustration of Geometric Relations in a Triangle with Isosceles Triangl
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2017-07-05 10:15
TriangleVectorTransformationGeometric ConstructionSlider Animation
This GeoGebra example dynamically illustrates a relationship that exists within a triangle when isosceles triangles are constructed outward on each of its sides. The core mathematical concepts involve properties of isosceles triangles, internal and external angles, rotation and reflection transformations, and the use of vectors in geometric proofs. The construction uses GeoGebra commands such as Segment, Rotate, Angle, Ray, Point, Min, Intersect, Circle, Dilate, Polygon, CircleSector, Vector, Translate, Line, and Midpoint to build the original triangle and the attached isosceles triangles, with sliders or dynamic points allowing real-time exploration of the relationships. It is suitable for high school and above students to investigate geometric transformations and vector applications, and can serve as a classroom demonstration or a self-guided exploration tool.

GeoGebra Commands & Parameters

  1. Segment(A, B) → f
  2. Rotate(B, (a * π / 180), A) → B'
  3. Angle(B, A, B') → α
  4. Ray(A, B') → g
  5. Point(g) → C
  6. Segment(B, C) → h
  7. Angle(C, B, A) → β
  8. Min(α, β) → b
  9. Rotate(B, c, A) → B'_1
  10. Angle(B, A, B'_1) → γ
  11. Ray(A, B'_1) → i
  12. Rotate(A, (-c), B) → A'
  13. Angle(A', B, A) → δ
  14. Ray(B, A') → j
  15. Intersect(i, j) → D
  16. Rotate(C, c, A) → C'
  17. Angle(C, A, C') → ε
  18. Rotate(A, (-c), C) → A'_1
  19. Angle(A'_1, C, A) → ζ
  20. Ray(A, C') → k
  21. Ray(C, A'_1) → l
  22. Intersect(l, k) → E
  23. Rotate(C, (-c), B) → C'_1
  24. Angle(C'_1, B, C) → η
  25. Rotate(B, c, C) → B'_2
  26. Angle(B, C, B'_2) → θ_1
  27. Ray(C, B'_2) → m
  28. Ray(B, C'_1) → n
  29. Intersect(m, n) → F
  30. Segment(A, D) → p
  31. Segment(D, B) → q
  32. Min({p, q, f}) → d
  33. Circle(A, (0.3 * d)) → e
  34. Intersect(e, f, 1) → G
  35. Intersect(i, p) → H
  36. Intersect(i, p) → I
  37. Intersect(i, p) → J
  38. Intersect(i, p) → K
  39. Intersect(e, i, 1) → L
  40. Segment(A, C) → r
  41. Dilate(A, If[0 < o ≤ 1, 1 - o, 0], D) → A'_2
  42. Dilate(B, If[0 < o ≤ 1, 1 - o, 0], D) → B'_3
  43. Dilate(A, If[1 < o ≤ 2, 2 - o, 0], E) → A'_3
  44. Dilate(C, If[1 < o ≤ 2, 2 - o, 0], E) → C'_2
  45. Dilate(B, If[2 < o ≤ 3, 3 - o, 0], F) → B'_4
  46. Dilate(C, If[2 < o ≤ 3, 3 - o, 0], F) → C'_3
  47. Segment(A, A'_2) → s
  48. Segment(B, B'_3) → t
  49. Segment(A, A'_3) → f_1
  50. Segment(C, C'_2) → g_1
  51. Segment(B, B'_4) → h_1
  52. Segment(C, C'_3) → i_1
  53. Polygon(A, B, C) → poly1, c_1, a_1, b_1
  54. Rotate(G, If[3 < o ≤ 4, ((o - 3) * γ), γ], A) → G'
  55. CircleSector(A, G, G') → d_1
  56. Dilate(A, If[4 < o ≤ 5, 5 - o, 0], B) → A'_4
  57. Vector(A, A'_4) → u
  58. Translate(d_1, u) → e_1
  59. Line(B, r) → j_1
  60. Line(B, s) → k_1
  61. Point(k_1) → M
  62. Rotate(e_1, If[5 < o ≤ 6, ((o - 5) * (π - γ)), π - γ], B) → p_1
  63. Dilate(B, If[6 < o ≤ 7, 7 - o, 0], C) → B'_5
  64. Vector(B, B'_5) → v
  65. Translate(p_1, v) → q_1
  66. Line(C, t) → l_1
  67. Point(l_1) → N
  68. Angle(N, C, E) → θ
  69. Rotate(q_1, If[7 < o ≤ 8, ((o - 7) * θ), θ], C) → r_1
  70. Midpoint(h) → O
  71. Midpoint(r) → P
  72. Dilate(B, If[8 < o ≤ 9, 9 - o, 0], O) → B'_6
  73. Vector(B, B'_6) → w
  74. Translate(p_1, w) → s_1
  75. Rotate(s_1, If[9 < o ≤ 10, ((-(o - 9)) * β), (-β)], O) → t_1
  76. Dilate(C, If[8 < o ≤ 9, 9 - o, 0], P) → C'_4
  77. Vector(C, C'_4) → u_1
  78. Translate(r_1, u_1) → c_2
  79. Rotate(c_2, If[9 < o ≤ 10, ((-(o - 9)) * (π - γ)), γ - π], P) → d_2
  80. Dilate(P, If[10 < o ≤ 11, 11 - o, 0], A) → P'
  81. Dilate(O, If[10 < o ≤ 11, 11 - o, 0], B) → O'
  82. Vector(P, P') → v_1
  83. Vector(O, O') → w_1
  84. Translate(d_2, v_1) → e_2
  85. Translate(t_1, w_1) → f_2
  86. Dilate(F, If[11 < o ≤ 12, 12 - o, 0], D) → F'
  87. Vector(F, F') → m_1
  88. Translate(C, m_1) → C'_5
  89. Translate(C'_3, m_1) → Q
  90. Segment(C'_5, Q) → n_1
  91. Dilate(E, If[12 < o ≤ 13, 13 - o, 0], D) → E'
  92. Vector(E, E') → u_2
  93. Translate(C, u_2) → C'_6
  94. Translate(C'_2, u_2) → R
  95. Segment(C'_6, R) → g_2