The Jerabek hyperbola is a circumconic that is the isogonal conjugate
of the Euler line (Kimberling 1998,
p. 237). Since it is a circumconic
passing through the orthocenter,
it is a rectangular hyperbola
and has center on the nine-point
circle. Its circumconic parameters are given by
![x:y:z=a[sin(2B)-sin(2C)]:b[sin(2C)-sin(2A)]:c[sin(2A)-sin(2B)],](/images/equations/JerabekHyperbola/NumberedEquation1.gif) |
(1)
|
meaning it has trilinear equation
![(a[sin(2B)-sin(2C)])/alpha+(b[sin(2C)-sin(2A)])/beta+(c[sin(2A)-sin(2B)])/gamma=0,](/images/equations/JerabekHyperbola/NumberedEquation2.gif) |
(2)
|
or equivalently
 |
(3)
|
(P. Moses, pers. comm., Apr. 19, 2005), where  ,  , and  are Conway triangle notation.
It passes through the vertices of a triangle as well as Kimberling centers  for  (circumcenter), 4 (orthocenter),
6 (symmedian point), 54 (Kosnita point), 64 isogonal conjugate of the de Longchamps point), 65 (orthocenter
of the contact triangle), 66
(isogonal conjugate of the
Exeter point), 67 (isogonal conjugate of the far-out
point), 68 (Prasolov point),
69, 70, 71, 72, 73, 74, 248, 265, 290, 695, 879, 895, 1173, 1175, 1176, 1177, 1242,
1243, 1244, 1245, 1246, 1439, 1798, 1903, 1942, 1987, 2213, 2435, 2574, 2575, 2992,
and 2993.
The Jerabek center is Kimberling center  , which has
equivalent triangle center
functions
(Kimberling 1998, p. 87).
Casey, J. A Treatise on the Analytical Geometry of the Point, Line, Circle,
and Conic Sections, Containing an Account of Its Most Recent Extensions with Numerous
Examples, 2nd rev. enl. ed. Dublin: Hodges, Figgis, & Co., pp. 448-451,
1893.
Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129,
1-295, 1998.
Pinkernell, G. M. "Cubic Curves in the Triangle Plane." J. Geom. 55,
141-161, 1996.
Vandeghen, A. "Some Remarks on the Isogonal and Cevian Transforms. Alignments of Remarkable Points of a Triangle." Amer. Math. Monthly 72, 1091-1094,
1965.
|
Contact the MathWorld Team
