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# Circumhyperbola

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

Circumcircle, Circumconic, Feuerbach's Conic Theorem, Rectangular Hyperbola

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## References

Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 198, 1959.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.

Circumhyperbola

## Cite this as:

Weisstein, Eric W. "Circumhyperbola." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Circumhyperbola.html