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
![alpha=((b^2rS_B-c^2qS_C)[a^2(q-r)+p(S_C-S_B)])/a,](/images/equations/Circumhyperbola/NumberedEquation1.svg) |
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 |
