Jerabek Hyperbola


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


meaning it has trilinear equation


or equivalently


(P. Moses, pers. comm., Apr. 19, 2005), where S_A, S_B, and S_C are Conway triangle notation.

It passes through the vertices of a triangle as well as Kimberling centers X_i for i=3 (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 X_(125), which has equivalent triangle center functions


(Kimberling 1998, p. 87).

See also

Circumcenter, de Longchamps Point, Euler Line, Isogonal Conjugate, Jerabek Antipode, Jerabek Center, Symmedian Point, Nine-Point Center, Orthocenter

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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.

Referenced on Wolfram|Alpha

Jerabek Hyperbola

Cite this as:

Weisstein, Eric W. "Jerabek Hyperbola." From MathWorld--A Wolfram Web Resource.

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