MathPlotAI Logo MathPlotAI

Equilateral Triangle Dynamic Property Demo

Equilateral Triangle Dynamic Property Demo
Loading the math board and drawing, please wait…
2016-08-29 03:47
TrianglePolygonTransformationGeometric Construction
This GeoGebra example dynamically illustrates a geometric property that holds exclusively for equilateral triangles. It explores invariance under rotation, dilation, and symmetry, leveraging core concepts such as equilateral triangles, geometric transformations, incircles, and circumcircles. Construction utilizes Polygon, Circle, Intersect, Rotate, Vector, Centroid, Dilate, and CircleArc commands to dynamically link points, segments, and arcs. Ideal for high school geometry, mathematical enrichment, and interactive GeoGebra learning, it helps students visualize and understand the unique behavior of equilateral triangles under dynamic manipulation.

GeoGebra Commands & Parameters

  1. Polygon(A, B, 3) → poly1, a, b, c, C
  2. Dilate(A, If[0 < d ≤ 1, 1 - d, 0], B) → A'
  3. Segment(A, A') → e
  4. Dilate(B, If[1 < d ≤ 2, 2 - d, 0], C) → B'
  5. Segment(B, B') → f
  6. Dilate(C, If[2 < d ≤ 3, 3 - d, 0], A) → C'
  7. Segment(C, C') → g
  8. Segment(A, B) → h
  9. Segment(A, C) → i
  10. Segment(B, C) → j
  11. Circle(A, (0.15 * h)) → k
  12. Intersect(k, i, 1) → D
  13. Intersect(k, h, 1) → E
  14. Rotate(A, If[3.5 < d ≤ 4.5, ((d - 3.5) * π / 3), 1.0471975511965976], A) → A'_1
  15. Rotate(A', If[3.5 < d ≤ 4.5, ((d - 3.5) * π / 3), 1.0471975511965976], A) → A''
  16. Segment(A'_1, A'') → e'
  17. Rotate(A, If[4.5 < d ≤ 5.5, ((-(d - 4.5)) * π / 3), -1.0471975511965976], B) → A'_2
  18. Rotate(A', If[4.5 < d ≤ 5.5, ((-(d - 4.5)) * π / 3), -1.0471975511965976], B) → A''_1
  19. Segment(A'_2, A''_1) → e'_1
  20. Dilate(A, If[6 < d ≤ 7, 7 - d, 0], B) → A'_3
  21. Vector(A, A'_3) → u
  22. Circle(B, (0.15 * h)) → r
  23. Intersect(f, e'_1) → G
  24. Intersect(f, e'_1) → H
  25. Intersect(f, e'_1) → I
  26. Intersect(f, e'_1) → J
  27. Intersect(r, j, 1) → F
  28. Intersect(r, h, 1) → K
  29. Dilate(B, If[8 < d ≤ 9, 9 - d, 0], C) → B'_1
  30. Vector(B, B'_1) → v
  31. Centroid(poly1) → L
  32. Circle(L, B) → p
  33. CircleArc(L, B, A) → q
  34. CircleArc(L, A, B) → t
  35. Point(t) → M
  36. Rotate(B, If[5.5 < d ≤ 6.5, (((d - 5.5) * 4) * π / 3), (4 * π / 3)], L) → B'_2
  37. CircleArc(L, B, B'_2) → c_1
  38. Rotate(A, If[6.5 < d ≤ 7.5, (((d - 6.5) * 2) * π / 3), (2 * π / 3)], L) → A'_4
  39. CircleArc(L, A, A'_4) → d_1
  40. Segment(A, M) → l
  41. Segment(B, M) → m
  42. Segment(C, M) → n
  43. Circle(M, l) → s
  44. Intersect(s, n) → N,
  45. Dilate(A, If[7.5 < d ≤ 8.5, 8.5 - d, 0], M) → A'_5
  46. Segment(A, A'_5) → f_1
  47. Dilate(B, If[8.5 < d ≤ 9.5, 9.5 - d, 0], M) → B'_3
  48. Segment(B, B'_3) → g_1
  49. Dilate(C, If[9.5 < d ≤ 10.5, 10.5 - d, 0], N) → C'_1
  50. Segment(C, C'_1) → h_1
  51. Dilate(N, If[10.5 < d ≤ 11.5, 11.5 - d, 0], M) → N'
  52. Segment(N, N') → i_1
  53. Circle(L, B) → e_1
  54. Rotate(A, If[11.5 < d ≤ 12.5, ((-(d - 11.5)) * π / 3), ((-π)) / 3], M) → A'_6
  55. Rotate(A'_5, If[11.5 < d ≤ 12.5, ((-(d - 11.5)) * π / 3), ((-π)) / 3], M) → O
  56. Segment(A'_6, O) → j_1
  57. Segment(A, M) → k_1
  58. Dilate(M, If[12.5 < d ≤ 13.5, 13.5 - d, 0], N) → M'
  59. Vector(M, M') → w
  60. Translate(B, w) → B'_4
  61. Translate(B'_3, w) → P
  62. Segment(B'_4, P) → l_1
  63. Segment(B, M) → m_1
  64. Rotate(B'_4, If[13.5 < d ≤ 14.5, ((d - 13.5) * π / 3), π / 3], N) → Q
  65. Rotate(P, If[13.5 < d ≤ 14.5, ((d - 13.5) * π / 3), π / 3], N) → P'
  66. Segment(Q, P') → n_1