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GoGeometry Action 93! Dynamic Geometry Problem

GoGeometry Action 93! Dynamic Geometry Problem
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2017-09-01 09:33
TriangleAngleTransformationGeometric ConstructionSlider Animation
This example is a dynamic and modifiable illustration of a classic geometry problem posted by Antonio Gutierrez (GoGeometry) on Twitter. It demonstrates a complex configuration involving angles, triangles, and various geometric transformations through an interactive GeoGebra construction. The core mathematical concepts include angle relationships, triangle properties, rotations, translations, dilations, and the combination of polygons and circles. The construction summary reveals the use of GeoGebra commands such as Rotate, Angle, Segment, Ray, Intersect, Circle, Min, CircleSector, Polygon, Dilate, Vector, Translate, and Line, along with numerous points, lines, circles, and polygon objects. This resource is suitable for high school or early college students to explore dynamic geometric relationships, and for teachers to use as a classroom demonstration to illustrate how transformations and angle interactions work together.

GeoGebra Commands & Parameters

  1. Rotate(B, (a * π / 180), A) → B'
  2. Angle(B, A, B') → α
  3. Segment(A, B) → f
  4. Ray(A, B') → g
  5. Rotate(A, (-(((2 * a) * π / 180))), B) → A'
  6. Angle(A', B, A) → β
  7. Ray(B, A') → h
  8. Intersect(g, h) → C
  9. Ray(A, B) → j
  10. Segment(A, C) → k
  11. Circle(C, 2) → c
  12. Intersect(c, g, 2) → D
  13. Angle(B, C, D) → γ
  14. Rotate(B, γ / 2, C) → B'_1
  15. Angle(B, C, B'_1) → δ
  16. Ray(C, B'_1) → i
  17. Intersect(j, i) → E
  18. Segment(B, E) → l
  19. Segment(C, B) → m
  20. Min({f, m, k}) → d
  21. Circle(A, (0.35 * d)) → e
  22. Circle(B, (0.35 * d)) → p
  23. Rotate(C, β / 2, B) → C'
  24. Angle(C, B, C') → ε
  25. Ray(B, C') → n
  26. Intersect(p, n, 1) → F
  27. Segment(B, F) → q
  28. Intersect(e, f, 1) → G
  29. Intersect(g, k) → H
  30. Intersect(g, k) → I
  31. Intersect(e, k) → J,
  32. CircleSector(A, G, J) → r
  33. Polygon(A, B, C) → poly1, c_1, a_1, b_1
  34. Dilate(A, If[0 < o ≤ 1, 1 - o, 0], C) → A'_1
  35. Dilate(A, If[0 < o ≤ 1, 1 - o, 0], B) → A'_2
  36. Segment(A, A'_2) → s
  37. Segment(A, A'_1) → t
  38. Dilate(B, If[1 < o ≤ 2, 2 - o, 0], C) → B'_2
  39. Segment(B, B'_2) → f_1
  40. Ray(C, D) → g_1
  41. Ray(A, B) → h_1
  42. Intersect(p, h_1) → K, L
  43. Circle(C, (0.35 * d)) → d_1
  44. Intersect(d_1, g_1) → M,
  45. Ray(C, B'_1) → i_1
  46. Intersect(d_1, i_1, 1) → N
  47. Dilate(A, If[3 < o ≤ 4, 4 - o, 0], B) → A'_3
  48. Vector(A, A'_3) → u
  49. Translate(r, u) → r'
  50. Rotate(r', If[4 < o ≤ 5, ((o - 4) * (π - α)), π - α], B) → r''
  51. Line(B, F) → j_1
  52. Intersect(t, j_1) → O
  53. Dilate(A, If[5 < o ≤ 6, 6 - o, 0], O) → A'_4
  54. Vector(A, A'_4) → v
  55. Translate(r, v) → r'_1
  56. Rotate(r'_1, If[6 < o ≤ 7, ((o - 6) * (π - (2 * α))), π - (2 * α)], O) → e_1
  57. Dilate(O, If[7 < o ≤ 8, 8 - o, 0], B) → O'
  58. Vector(O, O') → w
  59. Translate(e_1, w) → k_1
  60. Angle(N, C, M) → ζ
  61. Dilate(B, If[8 < o ≤ 9, 9 - o, 0], E) → B'_3
  62. Dilate(C, If[9 < o ≤ 10, 10 - o, 0], E) → C'_1
  63. Segment(C, C'_1) → m_1
  64. Line(A, B) → l_1
  65. Circle(B, C) → p_1
  66. Intersect(p_1, l_1, 2) → P
  67. Dilate(B, If[8 < o ≤ 9, 9 - o, 0], P) → B'_4
  68. Segment(B, B'_4) → n_1
  69. Dilate(P, If[9 < o ≤ 10, 10 - o, 0], E) → P'
  70. Segment(P, P') → q_1
  71. Rotate(B, If[11 < o ≤ 12, ((-(o - 11)) * (π - β)), π - β], B) → B'_5
  72. Rotate(B'_2, If[11 < o ≤ 12, ((-(o - 11)) * (π - β)), (-(π - β))], B) → Q
  73. Segment(B'_5, Q) → r_1
  74. Dilate(A, If[12 < o ≤ 13, 13 - o, 0], P) → A'_5
  75. Vector(A, A'_5) → u_1
  76. Translate(A, u_1) → A'_6
  77. Translate(A'_1, u_1) → R
  78. Segment(A'_6, R) → t'
  79. Rotate(A'_6, If[13 < o ≤ 14, ((-(o - 13)) * α), (-α)], P) → S
  80. Rotate(R, If[13 < o ≤ 14, ((-(o - 13)) * α), (-α)], P) → R'
  81. Segment(S, R') → t''
  82. CircleSector(C, N, M) → s_1
  83. Rotate(s_1, If[10 < o ≤ 11, ((-(o - 10)) * γ / 2), ((-γ)) / 2], C) → t_1

AI Drawing Prompt Reference

Draw a dynamic geometry figure: Start with point A, construct ray AB and ray AB' where AB' is obtained by rotating AB around A by angle a. These two rays intersect at point C, forming triangle ABC. Draw a small circle centered at C, intersecting ray AB' at D. From C, draw a ray making an angle with CB equal to half of angle BCD, intersecting AB at E. Draw three small circles centered at A, B, and C, each with radius 0.35 times the minimum side length of the triangle. At B, draw a ray making an angle with BC equal to half of angle ABC, intersecting the circle at B at F. At A, draw a circular arc connecting the intersections of the circle at A with AB and AC. Finally, use a slider to show dynamic scaled segments along the triangle sides, illustrating translation and scaling effects.
The prompt above was compiled from the original drawing commands and is for reference only.