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Four Thirteenths Triangle

Four Thirteenths Triangle
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2019-08-24 21:21
TriangleVectorGeometric Construction
In collections Exploring Feyman's Triangle
This example demonstrates the geometric construction of an inner triangle with an area ratio of four to thirteen compared to the outer triangle. Core math concepts include equilateral triangles, vectors, rotation, and area ratio calculations. Construction involves commands like Polygon, Midpoint, Segment, Ray, Intersect, Rotate, and Point in GeoGebra. Suitable for high school mathematics education to understand geometric transformations and area relationships.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C) → t1, c, a, b
  2. Midpoint(c) → X
  3. Midpoint(a) → Y
  4. Midpoint(b) → Z
  5. Segment(X, B) → f
  6. Segment(B, Y) → g
  7. Segment(Y, C) → h
  8. Segment(C, Z) → i
  9. Segment(Z, A) → j
  10. Segment(A, X) → k
  11. Midpoint(k) → G_1
  12. Midpoint(g) → I
  13. Midpoint(i) → K
  14. Ray(A, I) → l
  15. Ray(C, G_1) → m
  16. Ray(B, K) → n
  17. Intersect(l, m) → D
  18. Intersect(l, n) → E
  19. Intersect(m, n) → F
  20. Polygon(D, E, F) → t2, l_1, h_1, j_1
  21. Segment(B, F) → p
  22. Segment(A, E) → q
  23. Segment(C, D) → r
  24. Midpoint(j_1) → Q
  25. Midpoint(l_1) → G
  26. Midpoint(h_1) → M
  27. Polygon(B, Y, M) → t3, m_1, b_1, y_1
  28. Polygon(C, Q, Z) → t4, z_1, c_1, h_2
  29. Polygon(A, G, X) → t5, x_1, a_1, l_2
  30. Rotate(B, (-α), Y) → B'
  31. Rotate(Y, (-α), Y) → Y'
  32. Rotate(M, (-α), Y) → H'
  33. Polygon(B', Y', H') → t3', m', b', y'
  34. Rotate(C, (-α), Z) → C'
  35. Rotate(Q, (-α), Z) → I'
  36. Rotate(Z, (-α), Z) → Z'
  37. Polygon(C', I', Z') → t4', z', c', h'
  38. Rotate(A, (-α), X) → A'
  39. Rotate(G, (-α), X) → G'
  40. Rotate(X, (-α), X) → X'
  41. Polygon(A', G', X') → t5', x', a', l'
  42. Midpoint(y') → H
  43. Midpoint(z') → J
  44. Midpoint(x') → L
  45. Point(l_1) → N
  46. Segment(N, H) → s
  47. Point(j_1) → P
  48. Segment(E, L) → e
  49. Segment(H, F) → f_1
  50. Segment(J, D) → g_1
  51. Intersect(z', h') → R
  52. Intersect(z', g_1) → O
  53. Point(l') → S
  54. Polygon(G', S, E, N) → q1, g', s_1, e_1, n_1
  55. Intersect(m', y') → T
  56. Point(h_1) → U
  57. Polygon(T, F, U, H') → q3, t_1, f_2, u, i_1
  58. Segment(U, O) → t
  59. Polygon(A, I', Q, D) → q2, a_2, i', q_1, d
  60. Segment(Q, L) → k_1