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The Jarebek Hyperbola - Isogonal Conjugate of the Euler Line

The Jarebek Hyperbola - Isogonal Conjugate of the Euler Line
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2015-02-15 20:08
Coordinate Geometry
In collections A Tour of Triangle Geometry
Explore the nature of the Jarebek Hyperbola, the isogonal conjugate of the Euler Line, by dragging point P along the Euler Line. The Jarebek Hyperbola passes through specific points. Key math concepts include isogonal conjugate, Euler line, and hyperbola. Construction involves commands like Line, Circle, Center, OrthogonalLine, Intersect, Mirror, Midpoint, Segment, Point, Locus, and Polygon in GeoGebra. Suitable for high school students and those interested in geometric constructions.

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. OrthogonalLine(C, a) → e
  7. OrthogonalLine(A, b) → f
  8. Intersect(e, f) → H
  9. Line(H, O) → g
  10. Mirror(H, O) → L
  11. Midpoint(B, A) → D
  12. Segment(C, D) → h
  13. Intersect(g, h) → G
  14. Midpoint(B, C) → E
  15. Midpoint(C, A) → F
  16. Circle(D, F, E) → k
  17. Center(k) → N
  18. Point(g) → P
  19. Mirror(P, c) → P'_3
  20. Mirror(P, b) → P'_1
  21. Mirror(P, a) → P'_2
  22. Circle(P'_3, P'_2, P'_1) → p
  23. Center(p) → P'
  24. Locus(P', P) → loc1
  25. Polygon(A, B, C) → poly1, c_1, a_1, b_1
  26. Intersect(d, g) → J', J
  27. OrthogonalLine(J, c) → i
  28. OrthogonalLine(J, b) → j
  29. Intersect(i, c) → K
  30. Intersect(b, j) → M
  31. Line(K, M) → l
  32. OrthogonalLine(J', a) → m
  33. OrthogonalLine(J', b) → n
  34. Intersect(b, n) → I
  35. Intersect(a, m) → Q
  36. Line(I, Q) → q

AI Drawing Prompt Reference

Drag point P along the Euler line to construct the Jarebek hyperbola, the isogonal conjugate of the Euler line. Key points include H, O, G on the Euler line, and P' obtained by reflecting P. The hyperbola passes through points like J, K, M, I, Q. The construction involves the Euler line, reflections, circles, and intersections to obtain the hyperbola's locus.
The prompt above was compiled from the original drawing commands and is for reference only.