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Isogonal Conjugates: Euler Line and Jerabek Hyperbola

Isogonal Conjugates: Euler Line and Jerabek Hyperbola
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2019-08-04 09:12
TriangleCoordinate GeometryGeometric Construction
This example demonstrates the geometric relationship between isogonal conjugates, the Euler line, and the Jerabek hyperbola. Key mathematical concepts include isogonal conjugates, Euler line, Jerabek hyperbola, triangle centers (centroid, orthocenter, circumcenter), angle bisectors, reflections, unit circle, and coordinate geometry. The construction involves commands like Line, Polygon, Circle, Center, OrthogonalLine, Intersect, Point, AngularBisector, Mirror, Centroid, Midpoint, Conic, and Asymptote. Suitable for high school and advanced learners to explore properties and interrelations of special geometric figures.

GeoGebra Commands & Parameters

  1. Line(A, B) → f
  2. Line(B, C) → g
  3. Line(C, A) → h
  4. Polygon(A, B, C) → t1, c, a, b
  5. Circle(A, B, C) → d
  6. Center(d) → O
  7. OrthogonalLine(A, g) → i
  8. OrthogonalLine(B, h) → j
  9. Intersect(i, j) → H
  10. Line(H, O) → k
  11. Point(k) → P
  12. AngularBisector(B, A, C) → l
  13. AngularBisector(A, C, B) → m
  14. Line(A, P) → n
  15. Line(C, P) → p
  16. Mirror(n, l) → n'
  17. Mirror(p, m) → p'
  18. Intersect(n', p') → P'
  19. Centroid(t1) → G
  20. Midpoint(H, O) → N
  21. Midpoint(B, A) → D
  22. Midpoint(A, C) → E
  23. Midpoint(C, B) → F
  24. Circle(D, E, F) → e
  25. Conic(B, H, O, C, A) → q
  26. Asymptote(q) → r, s
  27. Intersect(e, r, 1) → I