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Circumhyperbola


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

 alpha=((b^2rS_B-c^2qS_C)[a^2(q-r)+p(S_C-S_B)])/a,

and the hyperbola itself given in trilinear coordinates by

 a(b^2rS_B-c^2qS_C)betagamma+b(c^2pS_C-a^2rS_A)gammaalpha 
 +c(a^2qS_A-b^2rS_B)alphabeta=0

(P. Moses, pers. comm., Jan. 27, 2005). The following tables summarizes a few such hyperbolas.

PP^'centerhyperbola
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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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.

Referenced on Wolfram|Alpha

Circumhyperbola

Cite this as:

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

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