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Conway Circle & Reuleaux Triangle: A Constant-Width Curve Through Six Points

Conway Circle & Reuleaux Triangle: A Constant-Width Curve Through Six Points
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2020-04-12 21:32
TriangleCircleTransformationGeometric ConstructionLocus
① This example demonstrates the relationship between the Conway circle and the Reuleaux triangle: the Conway circle passes through six specific points on the extended sides of a triangle, and a generalized Reuleaux triangle (a curve of constant width) also passes through the same six points. ② Core mathematical concepts include: incircle, circumcircle, constant-width curves, geometric construction, rotation, and symmetry. ③ Construction highlights: using GeoGebra commands such as Polygon, Circle, Line, Intersect, Segment, CircleArc, Rotate, Center, and OrthogonalLine, the construction involves a triangle ABC, its incircle c, and a Reuleaux triangle generated via rotation and circular arcs. Objects include anim, A, B, C, t1, c, a, b, d, e, f, g, h, i, D, E, F, G, H, I, j, k, l, m, n, p, q, r_1, s, t, c_1, d_1, e_1, f_1, g_1, r, h_1, r2, r3, a0. ④ Suitable for high school and above, this is ideal for exploring geometric constructions, properties of constant-width curves, and dynamic mathematics, serving as a demonstration or student inquiry tool.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C) → t1, c, a, b
  2. Circle(A, a) → d
  3. Circle(B, b) → e
  4. Circle(C, c) → f
  5. Line(A, B) → g
  6. Line(A, C) → h
  7. Line(C, B) → i
  8. Intersect(d, g, 1) → D
  9. Intersect(d, h, 1) → E
  10. Intersect(f, i, 1) → F
  11. Intersect(f, h, 2) → G
  12. Intersect(e, g, 2) → H
  13. Intersect(e, i, 2) → I
  14. Segment(C, B) → j
  15. Segment(A, D) → k
  16. Segment(A, E) → l
  17. Segment(B, A) → m
  18. Segment(C, F) → n
  19. Segment(C, G) → p
  20. Segment(A, C) → q
  21. Segment(B, I) → r_1
  22. Segment(B, H) → s
  23. Circle(D, E, I) → t
  24. CircleArc(A, E, D) → c_1
  25. CircleArc(A, G, H) → d_1
  26. CircleArc(B, D, F) → e_1
  27. CircleArc(C, F, G) → f_1
  28. CircleArc(B, H, I) → g_1
  29. Min(1, Max[anim - 1, 0]) → r
  30. CircleArc(C, I, E) → h_1
  31. Min(1, Max[anim - 2.2, 0]) → r2
  32. Min(1, Max[anim - 3.4, 0]) → r3
  33. Segment(E, G) → a0
  34. Rotate(E, (r * Angle[C, A, B]), A) → E'
  35. Rotate(G, (r * Angle[C, A, B]), A) → G'
  36. Segment(E', G') → a1
  37. Rotate(E', (r2 * Angle[A, B, C]), B) → E''
  38. Rotate(G', (r2 * Angle[A, B, C]), B) → G''
  39. Segment(E'', G'') → a2
  40. Rotate(E'', (r3 * Angle[B, C, A]), C) → E'''
  41. Rotate(G'', (r3 * Angle[B, C, A]), C) → G'''
  42. Segment(E''', G''') → a3
  43. Circle(E''', a) → k_1
  44. Circle(G''', b) → p_1
  45. Intersect(k_1, a4, 1) → J
  46. Intersect(p_1, a4, 1) → K
  47. Segment(E''', J) → i_1
  48. Segment(J, K) → j_1
  49. Segment(K, G''') → l_1
  50. CircleArc(A, E, E') → aa1
  51. CircleArc(A, G, G') → aa2
  52. CircleArc(B, D, E'') → aa3
  53. CircleArc(B, H, G'') → aa4
  54. CircleArc(C, F, E''') → aa5
  55. CircleArc(C, I, G''') → aa6
  56. Center(t) → L
  57. Min(1, Max[anim + 0.2, 0]) → r0
  58. Rotate(E, (r0 * 360°), L) → E'_1
  59. OrthogonalLine(L, g) → m_1
  60. Intersect(g, m_1) → M
  61. Circle(L, M) → q_1
  62. Segment(L, E'_1) → n_1
  63. Segment(L, E) → s_1
  64. Intersect(q_1, s_1) → N,
  65. Rotate(N, (r0 * 360°), L) → N'
  66. CircleArc(L, N, N') → ptc
  67. CircleArc(L, E, E'_1) → ptc2
  68. OrthogonalLine(G''', a4) → t_1
  69. OrthogonalLine(E''', a4) → a_1