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Morley's Equilateral Triangles

Morley's Equilateral Triangles
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2016-11-26 22:39
TriangleAngleGeometric Construction
This example demonstrates Morley's Trisector Theorem, where the intersection points of the angle trisectors of any triangle form an equilateral triangle. Key math concepts include equilateral triangles, angle bisectors, rotation, intersection, and the unit circle. Construction involves commands like Polygon, Angle, Rotate, Ray, Intersect, and Line. Suitable for high school geometry education and exploration by enthusiasts.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C) → poly1, c, a, b
  2. Angle(B, A, C) → α
  3. Angle(A, C, B) → β
  4. Angle(C, B, A) → γ
  5. Rotate(B, α / 3, A) → B'
  6. Angle(B, A, B') → δ
  7. Ray(A, B') → f
  8. Rotate(B', α / 3, A) → B''
  9. Angle(B', A, B'') → ε
  10. Ray(A, B'') → g
  11. Rotate(C, γ / 3, B) → C'
  12. Angle(C, B, C') → ζ
  13. Ray(B, C') → h
  14. Rotate(C', γ / 3, B) → C''
  15. Angle(C', B, C'') → η
  16. Ray(B, C'') → i
  17. Rotate(A, β / 3, C) → A'
  18. Angle(A, C, A') → θ
  19. Ray(C, A') → j
  20. Rotate(A', β / 3, C) → A''
  21. Angle(A', C, A'') → ι
  22. Ray(C, A'') → k
  23. Intersect(h, k) → D
  24. Intersect(f, i) → E
  25. Intersect(g, j) → F
  26. Polygon(D, E, F) → poly2, f_1, d, e
  27. Rotate(A, (180° - γ) / 3, B) → A'_1
  28. Angle(A, B, A'_1) → μ
  29. Line(B, A'_1) → n
  30. Rotate(A'_1, (180° - γ) / 3, B) → H
  31. Angle(A'_1, B, H) → ν
  32. Line(B, H) → p
  33. Rotate(B, (180° - β) / 3, C) → B'_3
  34. Angle(B, C, B'_3) → ξ
  35. Line(C, B'_3) → q
  36. Rotate(B'_3, (180° - β) / 3, C) → I
  37. Angle(B'_3, C, I) → ο
  38. Line(C, I) → r
  39. Rotate(C, (180° - α) / 3, A) → C'_1
  40. Angle(C, A, C'_1) → κ
  41. Line(A, C'_1) → l
  42. Rotate(C'_1, (180° - α) / 3, A) → G
  43. Angle(C'_1, A, G) → λ
  44. Line(A, G) → m
  45. Intersect(n, f) → J
  46. Intersect(r, g) → K
  47. Intersect(l, j) → L
  48. Intersect(p, k) → M
  49. Intersect(m, i) → N
  50. Intersect(q, h) → O
  51. Intersect(n, m) → P
  52. Intersect(p, q) → Q
  53. Intersect(r, l) → R
  54. Polygon(Q, K, J) → poly3, j_1, q_1, k_1
  55. Polygon(L, P, M) → poly4, m_1, l_1, p_1
  56. Polygon(N, R, O) → poly5, o, n_1, r_1
  57. Polygon(P, Q, R) → poly6, r_2, p_2, q_2