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More Properties of the Jerabek Hyperbola

More Properties of the Jerabek Hyperbola
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2015-02-16 08:28
TriangleCircleCoordinate GeometryGeometric Construction
In collections A Tour of Triangle Geometry
This example demonstrates the properties of the Jerabek hyperbola, which is the isogonal conjugate of the Euler line. By constructing the hyperbola, we can observe its intersection with the circumcircle at point E'. Using GeoGebra commands such as Line, Circle, Intersect, etc., we can explore the geometric properties of the hyperbola. Suitable for high school geometry education.

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_1
  15. Midpoint(C, A) → F
  16. Circle(D, F, E_1) → 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
  37. Mirror(g, a) → g'
  38. Mirror(g, c) → g'_1
  39. Mirror(g, b) → g'_2
  40. Intersect(g', g'_1) → E
  41. Mirror(E, O) → E'
  42. OrthogonalLine(B, c) → r
  43. Intersect(a, e) → R
  44. Intersect(k, f, 1) → S
  45. Intersect(c, r) → T
  46. Polygon(R, S, T) → poly2, t, r_1, s
  47. Intersect(l, q) → Center
  48. Tool1(A, R, T) → d_1
  49. Tool1(C, T, S) → e_1
  50. Tool1(S, R, B) → f_1

AI Drawing Prompt Reference

Draw a triangle and its circumcircle. Find the intersection of the perpendicular bisector of one side with the circumcircle. Use this point to draw a circle that intersects the other two sides. Connect these intersection points to the vertices of the triangle and draw perpendicular lines. Observe the intersections of these perpendiculars with the triangle's sides and connect them to form a new shape. Finally, draw a line from the midpoint of one side to the opposite vertex and its perpendicular line, noting the intersections with the previous shapes.
The prompt above was compiled from the original drawing commands and is for reference only.