A circumhyperbola is a circumconic that is a hyperbola.

A rectangular circumhyperbola always passes through the orthocenter H 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.

For a point P and its antipode P^' on the circumcircle, the Simson lines of P and P^' 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 PP^'. The center of this hyperbola for a trilinear point p:q:r 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.

X_(98)X_(99)X_(2679)through (4, 32, 237, 263, 511, 512, 2211, 2698)
X_(101)X_(103)X_(1566)through (4, 279, 514, 516, 2724)
X_(1113)X_(1114)X_(125)Jerabek hyperbola
X_(1379)X_(1380)X_(115)Kiepert hyperbola
X_(1381)X_(1382)X_(11)Feuerbach hyperbola

See also

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

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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.

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Cite this as:

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

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