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Kiepert Hyperbola: Dynamic Construction of a Triangle's Circumhyperbola

Kiepert Hyperbola: Dynamic Construction of a Triangle's Circumhyperbola
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2017-01-22 09:10
PolygonAngleTransformationGeometric ConstructionLocus
This example demonstrates the construction of the Kiepert hyperbola, a special hyperbola generated by rotating the sides of a triangle and connecting corresponding intersection points, with the final curve traced via the Locus command. The core mathematical concepts involve triangles, rotation, similarity transformations, intersection loci, and the geometric properties of hyperbolas. The construction uses GeoGebra commands such as Line, Rotate, Intersect, Segment, Angle, Polygon, and Locus; key objects include triangle vertices A, B, C, rotated lines f, g, h and their intersections X, Y, Z, as well as auxiliary points G, D, E. This is suitable for high school and above students or geometry enthusiasts, serving as an advanced dynamic geometry exploration of the Kiepert hyperbola and its relation to circumscribed geometry of a triangle.

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
  29. Polygon(Z, Y, X) → poly1, D, z_1, E
  30. Locus(G, α) → loc1
  31. Polygon(A, B, C) → poly2, c, a, b