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Dynamic Illustration of Special Relationships with Equilateral Triangles

Dynamic Illustration of Special Relationships with Equilateral Triangles
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2017-08-18 19:59
TrianglePolygonAngleTrigonometryGeometric Construction
This example dynamically illustrates a special relationship that emerges when two equilateral triangles are constructed on two sides of any triangle. It involves concepts like congruence, equilateral triangles, geometric distribution, logarithms, difference and slope, and trigonometry. The construction uses commands such as Segment, Rotate, Angle, Ray, Point, Polygon, Midpoint, and Dilate in GeoGebra. It is suitable for high school students and educators exploring geometric transformations and relationships.

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(C, A) → F
  24. Midpoint(B, C) → G
  25. Dilate(C, If[2 < d ≤ 3, 3 - d, 0], F) → C'_2
  26. Dilate(C, If[2 < d ≤ 3, 3 - d, 0], G) → C'_3
  27. Segment(C, C'_2) → q
  28. Segment(C, C'_3) → r
  29. Rotate(C, If[3 < d ≤ 4, ((d - 3) * π), π], F) → C'_4
  30. Rotate(C'_2, If[3 < d ≤ 4, ((d - 3) * π), π], F) → H
  31. Segment(C'_4, H) → q'
  32. Rotate(C, If[3 < d ≤ 4, ((-(d - 3)) * π), (-π)], G) → C'_5
  33. Rotate(C'_3, If[3 < d ≤ 4, ((-(d - 3)) * π), (-π)], G) → K
  34. Segment(C'_5, K) → r'
  35. Midpoint(E, D) → L
  36. Polygon(F, G, L) → poly4, l_1, f_1, g_1
  37. Dilate(D, If[4 < d ≤ 5, 5 - d, 0], L) → D'
  38. Dilate(L, If[5 < d ≤ 6, 6 - d, 0], E) → L'
  39. Segment(D, D') → s
  40. Segment(L, L') → t
  41. Rotate(D, If[6 < d ≤ 7, ((-(d - 6)) * π), (-π)], L) → D'_1
  42. Rotate(D', If[6 < d ≤ 7, ((-(d - 6)) * π), (-π)], L) → D''
  43. Segment(D'_1, D'') → s'
  44. Dilate(L, If[7 < d ≤ 8, 8 - d, 0], G) → L'_1
  45. Segment(L, L'_1) → e
  46. Rotate(L, If[8 < d ≤ 9, ((d - 8) * π / 3), π / 3], G) → L'_2
  47. Rotate(L'_1, If[8 < d ≤ 9, ((d - 8) * π / 3), π / 3], G) → M
  48. Segment(L'_2, M) → e'
  49. Rotate(L'_2, If[9 < d ≤ 10, ((d - 9) * π / 3), π / 3], F) → N
  50. Rotate(M, If[9 < d ≤ 10, ((d - 9) * π / 3), π / 3], F) → M'
  51. Segment(N, M') → e''