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Kiepert Hyperbola and Brocard Axis

Kiepert Hyperbola and Brocard Axis
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2015-02-16 08:39
TriangleGeometric ConstructionLocus
In collections Besondere Punkte/Special Points A Tour of Triangle Geometry
This example demonstrates the geometric relationship between the Kiepert hyperbola and the Brocard axis. Key mathematical concepts include isogonal conjugation, centroid, and symmedian point. Construction involves GeoGebra commands such as Line, Circle, Centroid, Mirror, etc. Suitable for high school students or geometry enthusiasts.

GeoGebra Commands & Parameters

  1. Line(A, B) → a
  2. Line(B, C) → b
  3. Line(C, A) → c
  4. Circle(A, B, C) → d
  5. Center(d) → O
  6. Centroid(Polygon[A, B, C]) → G
  7. Mirror(G, c) → G'
  8. Mirror(G, a) → G'_1
  9. Mirror(G, b) → G'_2
  10. LineBisector(G'_1, G') → e
  11. LineBisector(G', G'_2) → f
  12. Intersect(e, f) → K
  13. Line(K, O) → g
  14. Point(g) → P
  15. Mirror(P, a) → P'_3
  16. Mirror(P, c) → P'_1
  17. Mirror(P, b) → P'_2
  18. LineBisector(P'_3, P'_1) → h
  19. LineBisector(P'_1, P'_2) → i
  20. Intersect(h, i) → P'
  21. Locus(P', P) → loc1
  22. Polygon(A, B, C) → poly1, c_1, a_1, b_1

AI Drawing Prompt Reference

Construct a triangle, find its centroid, and reflect it over each side to get three points. Connect these points to the vertices to form two angle bisectors, intersecting at point K. Draw a circle centered at the centroid. Connect K to the center to form a line segment. Take a point P on this segment, reflect it over each side to get P1, P2, P3. Connect P1P3 and P1P2, intersecting at P'. Plot the locus of P and P', demonstrating the geometric relationship between the Kiepert hyperbola and Brocard axis.
The prompt above was compiled from the original drawing commands and is for reference only.