A circumhyperbola is a circumconic that is a hyperbola.
A rectangular circumhyperbola always passes through the orthocenter and has center on the nine-point circle (Kimberling 1998, p. 236), a result known as the Feuerbach's conic theorem (Coolidge 1959, p. 198).
The following table summarizes a number of rectangular circumhyperbolas, together with their centers and most important fifth incident point.
rectangular circumhyperbola | Kimberling | center | Kimberling | incident point(s) |
Feuerbach hyperbola | Feuerbach point | , , , | incenter, Gergonne point, Nagel point, mittenpunkt | |
Jerabek hyperbola | , | circumcenter, symmedian point | ||
Kiepert hyperbola | , , , | triangle centroid, Spieker center, first Fermat point, second Fermat point | ||
nine-point center |
For a point and its antipode on the circumcircle, the Simson lines of and meet at a point on the nine-point circle. Furthermore, this point is the center of the rectangular circumhyperbola that is the isogonal conjugate of the line . The center of this hyperbola for a trilinear point has center function
and the hyperbola itself given in trilinear coordinates by
(P. Moses, pers. comm., Jan. 27, 2005). The following tables summarizes a few such hyperbolas.
center | hyperbola | ||
through (4, 32, 237, 263, 511, 512, 2211, 2698) | |||
through (4, 279, 514, 516, 2724) | |||
Jerabek hyperbola | |||
Kiepert hyperbola | |||
Feuerbach hyperbola |