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Circles Through Fermat Point & Kiepert Hyperbola

Circles Through Fermat Point & Kiepert Hyperbola
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2017-01-23 04:45
TriangleCircleGeometric ConstructionLocus
This example demonstrates the geometric relationship between circles passing through the Fermat point of a triangle and the Kiepert hyperbola, showing the intersection loci of these circles with the hyperbola. Core mathematical concepts include the Fermat point, Kiepert hyperbola, equilateral triangles, centroid, and the intersection properties of circles and hyperbolas. Construction highlights: using GeoGebra commands such as Line, Rotate, Intersect, Segment, Angle, Polygon, Locus, OrthogonalLine, Centroid, Circle, Center, LineBisector, and Point, the construction builds triangles, equilateral triangles, the Fermat point, the Kiepert hyperbola, and circles through the Fermat point, with Locus generating the trajectory. Suitable for high school and above students or geometry enthusiasts, this is useful for exploring connections between special triangle points and conic sections, and can serve as a dynamic geometry teaching demonstration.

GeoGebra Commands & Parameters

  1. Line(A, B) → f
  2. Line(B, C) → g
  3. Line(C, A) → h
  4. Rotate(h, (-α), C) → h'
  5. Rotate(h, α, A) → h'_1
  6. Intersect(h', h'_1) → Y
  7. Rotate(g, α, C) → g'
  8. Rotate(g, (-α), B) → g'_1
  9. Intersect(g', g'_1) → X
  10. Rotate(f, α, B) → f'
  11. Rotate(f, (-α), A) → f'_1
  12. Intersect(f', f'_1) → Z
  13. Segment(B, Z) → i
  14. Segment(Z, A) → j
  15. Segment(A, Y) → k
  16. Segment(Y, C) → l
  17. Segment(C, X) → m
  18. Segment(X, B) → n
  19. Angle(A, B, Z) → β
  20. Angle(Z, A, B) → γ
  21. Angle(C, A, Y) → δ
  22. Angle(Y, C, A) → ε
  23. Angle(X, B, C) → ζ
  24. Angle(B, C, X) → η
  25. Line(Z, C) → p
  26. Line(X, A) → q
  27. Line(B, Y) → r
  28. Intersect(p, q) → G_1
  29. Polygon(Z, Y, X) → poly1, D, z_1, E
  30. Locus(G_1, α) → loc1
  31. Polygon(A, B, C) → poly2, c, a, b
  32. Polygon(C, B, 3) → poly3, s, t, d, F
  33. Polygon(A, C, 3) → poly4, e, f_1, g_1, H_1
  34. Line(B, H_1) → h_1
  35. Line(A, F) → i_1
  36. Intersect(h_1, i_1) → I
  37. Polygon(B, C, 3) → poly5, j_1, k_1, l_1, J
  38. Polygon(A, B, 3) → poly6, m_1, n_1, p_1, K
  39. Line(K, C) → q_1
  40. Line(J, A) → r_1
  41. Intersect(q_1, r_1) → L
  42. OrthogonalLine(A, g) → s_1
  43. OrthogonalLine(C, f) → t_1
  44. Intersect(s_1, t_1) → H
  45. Centroid(poly1) → G
  46. Line(H, G) → a_1
  47. Circle(A, B, C) → c_1
  48. Center(c_1) → O
  49. LineBisector(L, I) → b_1
  50. Point(b_1) → M
  51. Circle(M, L) → d_1
  52. OrthogonalLine(M, a_1) → e_1
  53. Intersect(d_1, e_1) → N, P
  54. Segment(P, N) → f_2