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Center of the Kiepert Hyperbola

Center of the Kiepert Hyperbola
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2015-02-16 08:47
TriangleCoordinate GeometryGeometric Construction
In collections A Tour of Triangle Geometry
This example demonstrates how to find the center of the Kiepert hyperbola. First, construct the orthic triangle DEF of triangle ABC. Then, find the Brocard axes of triangles ADE, BDF, and CEF. Through these steps, the center of the Kiepert hyperbola can be determined. The example involves concepts such as triangles, orthic triangles, and Brocard axes. It uses GeoGebra commands such as Line, Circle, Center, Centroid, Mirror, LineBisector, Intersect, Point, Locus, Polygon, and OrthogonalLine. Suitable for high school geometry education and learning.

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
  23. OrthogonalLine(A, b) → j
  24. OrthogonalLine(B, c) → k
  25. OrthogonalLine(C, a) → l
  26. Intersect(a, l) → D
  27. Intersect(c, k) → E
  28. Intersect(b, j) → F
  29. Polygon(D, E, F) → poly2, f_1, d_1, e_1
  30. Intersect(l, j) → H
  31. Tool2(A, D, E) → m
  32. Tool2(B, D, F) → n
  33. Tool2(F, E, C) → p
  34. Intersect(m, n) → K_1

AI Drawing Prompt Reference

Draw the circumcircle of triangle ABC and find its center O. Construct the orthocenter G of triangle ABC, then reflect G over BC, AC, and AB to get G', G'_1, and G'_2. Draw the angle bisectors e and f of G'_1 and G', G' and G'_2, respectively, and their intersection K is the center of the Kiepert hyperbola. Draw the line g through K and O, reflect points on g over BC, AC, and AB to get P'_3, P'_1, and P'_2, then draw the angle bisectors h and i of P'_3 and P'_1, P'_1 and P'_2, respectively, and their intersection P' is another focus of the Kiepert hyperbola. The locus of P' is the Kiepert hyperbola. Construct the orthic triangle DEF of triangle ABC, draw the perpendiculars from D, E, and F to the opposite sides to get H, then construct the orthic triangle of DEF and find its orthocenter K_1.
The prompt above was compiled from the original drawing commands and is for reference only.