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Dynamic Construction of Equilateral Triangles on Arbitrary Sides

Dynamic Construction of Equilateral Triangles on Arbitrary Sides
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2019-10-17 08:28
TrianglePolygonVectorParametric EquationGeometric Construction
This example dynamically illustrates the construction and properties of equilateral triangles built on the sides of any polygon. It involves core mathematical concepts such as equilateral triangle construction, rotation, vectors, parametric curves, and geometric transformations. The construction utilizes commands like Segment, Rotate, Angle, Point, Polygon, Midpoint, Dilate, Vector, Translate, and Line in GeoGebra. Suitable for high school and advanced mathematics education and geometric exploration.

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. Segment(A, C) → i
  8. Polygon(A, B, C) → poly1, c, a_1, b
  9. Polygon(C, B, 3) → poly2, j, k, l, D
  10. Polygon(A, C, 3) → poly3, m, n, p, E
  11. Rotate(A, If[0 < d ≤ 1, (d * π / 3), π / 3], A) → A'
  12. Rotate(C, If[0 < d ≤ 1, (d * π / 3), π / 3], A) → C'
  13. Segment(A', C') → i'
  14. Rotate(B, If[0 < d ≤ 1, (d * π / 3), π / 3], C) → B'_1
  15. Rotate(C, If[0 < d ≤ 1, (d * π / 3), π / 3], C) → C'_1
  16. Segment(B'_1, C'_1) → h'
  17. Rotate(B'_1, If[1 < d ≤ 2, ((d - 1) * π / 3), π / 3], D) → I
  18. Rotate(C'_1, If[1 < d ≤ 2, ((d - 1) * π / 3), π / 3], D) → J
  19. Segment(I, J) → h''
  20. Rotate(A', If[1 < d ≤ 2, ((d - 1) * π / 3), π / 3], E) → A''
  21. Rotate(C', If[1 < d ≤ 2, ((d - 1) * π / 3), π / 3], E) → C''
  22. Segment(A'', C'') → i''
  23. Midpoint(E, D) → L
  24. Dilate(D, If[4 < d ≤ 5, 5 - d, 0], L) → D'
  25. Dilate(L, If[5 < d ≤ 6, 6 - d, 0], E) → L'
  26. Rotate(D, If[6 < d ≤ 7, ((-(d - 6)) * π), (-π)], L) → D'_1
  27. Rotate(D', If[6 < d ≤ 7, ((-(d - 6)) * π), (-π)], L) → D''
  28. Dilate(A, If[3 < d ≤ 4, 4 - d, 0], D) → A'_1
  29. Dilate(D, If[5 < d ≤ 6, 6 - d, 0], B) → D'_2
  30. Vector(D, D'_2) → u
  31. Translate(A, u) → A'_2
  32. Translate(A'_1, u) → F
  33. Segment(A'_2, F) → q'
  34. Line(E, B) → s
  35. Segment(A, B) → q
  36. Rotate(A, If[2 < d ≤ 3, ((d - 2) * π / 3), π / 3], A) → A'_3
  37. Rotate(B, If[2 < d ≤ 3, ((d - 2) * π / 3), π / 3], A) → B'_2
  38. Segment(A'_3, B'_2) → q'_1
  39. Rotate(A'_3, If[3 < d ≤ 4, ((d - 3) * π / 3), π / 3], B'_2) → G
  40. Rotate(B'_2, If[3 < d ≤ 4, ((d - 3) * π / 3), π / 3], B'_2) → H
  41. Segment(G, H) → r
  42. Polygon(A, B'_2, B) → t1, b_1, a_2, b'_2
  43. Dilate(C, If[5 < d ≤ 6, 6 - d, 0], E) → C'_2
  44. Vector(C, C'_2) → v
  45. Translate(B'_1, v) → K
  46. Translate(C'_1, v) → M
  47. Segment(K, M) → h''_1
  48. Dilate(C, If[6 < d ≤ 7, 7 - d, 0], D) → C'_3
  49. Vector(C, C'_3) → w
  50. Translate(A'', w) → A'''
  51. Translate(C'', w) → C'''
  52. Segment(A''', C''') → i'''
  53. Polygon(E, C, D, B'_2) → q1, e, c_1, d_1, b'_1