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Dynamic Geometry of a Special Isosceles Triangle (GoGeometry Action 43!)

Dynamic Geometry of a Special Isosceles Triangle (GoGeometry Action 43!)
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2017-05-19 00:01
TriangleAngleTrigonometryTransformationGeometric Construction
This example dynamically illustrates a geometry problem involving a special isosceles triangle, inspired by a tweet from Antonio Gutierrez. The core mathematical concepts include properties of isosceles triangles, parallelograms, rotation and dilation transformations, and trigonometric relationships among angles. The construction uses GeoGebra commands such as Rotate, Angle, Segment, Ray, Intersect, Circle, LineBisector, Dilate, Polygon, CircleSector, Midpoint, Vector, Translate, and CircleArc, creating numerous objects (points A, B, C, D, E, F, G, H, I, J, K, L, M, N, etc.), lines, circles, angle bisectors, dilated figures, and sectors. Dynamic sliders (α, β, γ, δ) allow interactive exploration of the configuration. It is suitable for high school or early college students studying geometric transformations and special triangle properties, and serves as an engaging classroom demonstration tool.

GeoGebra Commands & Parameters

  1. Rotate(B, (80 * π / 180), A) → B'
  2. Angle(B, A, B') → α
  3. Rotate(A, (-80 * π / 180), B) → A'
  4. Angle(A, B, A') → β
  5. Segment(A, B) → f
  6. Ray(A, B') → g
  7. Ray(B, A') → h
  8. Intersect(g, h) → C
  9. Segment(A, C) → i
  10. Segment(C, B) → j
  11. Rotate(B, π / 3, C) → B'_1
  12. Segment(B, B'_1) → k
  13. Segment(C, B'_1) → l
  14. Rotate(B, π / 9, C) → B'_2
  15. Angle(B, C, B'_2) → γ
  16. Rotate(B'_2, π / 9, C) → D
  17. Angle(B'_2, C, D) → δ
  18. Ray(C, B'_2) → m
  19. Ray(C, D) → n
  20. Intersect(k, m) → E
  21. Intersect(k, n) → F
  22. Circle(C, (0.2 * j)) → c
  23. Intersect(c, j, 1) → G
  24. Intersect(c, i, 1) → H
  25. Segment(C, E) → p
  26. Segment(C, F) → q
  27. Intersect(c, p, 1) → I
  28. Intersect(c, q, 1) → J
  29. Circle(B, A) → d
  30. Intersect(d, i) → K, L
  31. Intersect(d, j, 1) → M
  32. LineBisector(i) → r
  33. Intersect(r, j) → N
  34. Dilate(A, If[0 < a ≤ 1, 1 - a, 0], B) → A'_1
  35. Dilate(C, If[0 < a ≤ 1, 1 - a, 0], B) → H'
  36. Segment(C, H') → t
  37. Rotate(C, If[1 < a ≤ 2, (((-(a - 1)) * 20) * π / 180), (-20 * π / 180)], C) → C'
  38. Rotate(H', If[1 < a ≤ 2, (((-(a - 1)) * 20) * π / 180), (-20 * π / 180)], C) → H''
  39. Segment(C', H'') → t'
  40. Dilate(A, If[2 < a ≤ 3, 3 - a, 0], B) → A'_2
  41. Segment(A, A'_2) → s
  42. Polygon(A, B, C) → poly1, c_1, a_1, b
  43. Rotate(C, If[3 < a ≤ 4, ((a - 3) * π / 3), π / 3], C) → C'_1
  44. Rotate(H', If[3 < a ≤ 4, ((a - 3) * π / 3), π / 3], C) → H''_1
  45. Segment(C'_1, H''_1) → t'_1
  46. Rotate(C, If[4 < a ≤ 5, ((-(a - 4)) * π / 3), ((-π)) / 3], B) → C'_2
  47. Rotate(H', If[4 < a ≤ 5, ((-(a - 4)) * π / 3), ((-π)) / 3], B) → H''_2
  48. Segment(C'_2, H''_2) → t'_2
  49. Polygon(C, B, B'_1) → poly2, b'_1, c_2, b_1
  50. Intersect(c, l) → O,
  51. Dilate(C, If[5 < a ≤ 6, 6 - a, 0], E) → C'_3
  52. Dilate(C, If[5 < a ≤ 6, 6 - a, 0], F) → C'_4
  53. Segment(C, C'_3) → e
  54. Segment(C, C'_4) → f_1
  55. CircleSector(C, J, O) → d_1
  56. Rotate(d_1, If[7 < a ≤ 8, (((-(a - 7)) * 20) * π / 180), (-20 * π / 180)], C) → e_1
  57. Rotate(e_1, If[9 < a ≤ 10, (((-(a - 9)) * 20) * π / 180), (-20 * π / 180)], C) → g_1
  58. Rotate(g_1, If[11 < a ≤ 12, (((-(a - 11)) * 20) * π / 180), (-20 * π / 180)], C) → h_1
  59. Midpoint(C, A) → P
  60. Dilate(C, If[12.5 < a ≤ 13.5, 13.5 - a, 0], P) → C'_5
  61. Vector(C, C'_5) → u
  62. Translate(h_1, u) → k_1
  63. Rotate(k_1, If[13.5 < a ≤ 14.5, (((a - 13.5) * 160) * π / 180), (160 * π / 180)], P) → p_1
  64. Dilate(A, If[14.5 < a ≤ 15.5, 15.5 - a, 0], N) → A'_3
  65. Segment(A, A'_3) → i_1
  66. Dilate(P, If[15.5 < a ≤ 16.5, 16.5 - a, 0], A) → P'
  67. Vector(P, P') → v
  68. Translate(p_1, v) → q_1
  69. Rotate(A, If[16.5 < a ≤ 17.5, (((-(a - 16.5)) * 80) * π / 180), (-80 * π / 180)], B) → A'_4
  70. Segment(B, A'_4) → j_1
  71. CircleArc(B, A'_4, A) → r_1
  72. Angle(L, B, A) → ε
  73. Segment(B, L) → l_1
  74. Polygon(A, C, N) → poly3, n_1, a_2, c_3
  75. Polygon(A, N, B) → poly4, b_2, a_3, n_2
  76. Dilate(A, If[18.5 < a ≤ 19.5, 19.5 - a, 0], L) → A'_5
  77. Vector(A, A'_5) → w
  78. Segment(A, A'_5) → m_1
  79. Dilate(L, If[19.5 < a ≤ 20.5, 20.5 - a, 0], M) → L'
  80. Vector(L, L') → u_1
  81. Translate(A, u_1) → A'_6
  82. Translate(A'_5, u_1) → Q
  83. Segment(A'_6, Q) → s_1
  84. Rotate(A'_6, If[20.5 < a ≤ 21.5, (((-(a - 20.5)) * 160) * π / 180), (-160 * π / 180)], M) → R
  85. Rotate(Q, If[20.5 < a ≤ 21.5, (((-(a - 20.5)) * 160) * π / 180), (-160 * π / 180)], M) → Q'
  86. Segment(R, Q') → t_1

AI Drawing Prompt Reference

Draw a special isosceles triangle with a vertex angle of 80 degrees and base angles of 50 degrees. From the vertex, draw two rays extending to the base line, intersecting at a point beyond the base, forming a larger triangle. On one side of the base, construct an equilateral triangle, and from the base endpoint draw two rays inside it, intersecting the equilateral triangle's sides. Connect these intersection points to the base endpoint. With the midpoint of the base as center, draw a circle with radius a fraction of the base length, intersecting the triangle sides and internal segments; mark these intersections. Then, with the base endpoint as center and the base length as radius, draw another circle intersecting the triangle sides. Construct the perpendicular bisector of the base, intersecting the other leg. Finally, use a slider to control a sequence of rotations, dilations, and segment translations, dynamically showing transformations of the internal structure.
The prompt above was compiled from the original drawing commands and is for reference only.