TOPICS
Search

Feuerbach Hyperbola


FeuerbachHyperbola

A circumconic hyperbola, which therefore passes through the orthocenter, is a rectangular hyperbola, and has center on the nine-point circle. Its circumconic parameters are given by

 x:y:z=cosB-cosC:cosC-cosA:cosA-cosB

meaning it has trilinear equation

 (cosB-cosC)/alpha+(cosC-cosA)/beta+(cosA-cosB)/gamma=0

(Kimberling 1998, p. 237).

Its center is the Feuerbach point F (Kimberling 1998, p. 237).

It passes through the vertices of a triangle and Kimberling centers X_i for i=1 (incenter),

4 (orthocenter), 7 (Gergonne point), 8 (Nagel point), 9 (mittenpunkt), 21 (Schiffler point), 79, 80, 84, 90, 104, 177, 256, 294, 314, 885, 941, 943, 981, 983, 987, 989, 1000, 1039, 1041, 1061, 1063, 1156, 1172, 1251, 1320, 1389, 1392, 1476, 1896, 1937, 2298, 2320, 2335, 2344, 2346, 2481, 2648, and 2997.

The Feuerbach hyperbola is the isogonal conjugate of the line OI, where O is the circumcenter and I is the incenter of DeltaABC.


See also

Circumconic, Feuerbach's Conic Theorem, Kiepert Hyperbola, Jerabek Hyperbola, Stammler Hyperbola

Explore with Wolfram|Alpha

References

Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.Mandart H. "Sur l'hyperbole de Feuerbach." Mathesis, 81-89, 1893.Rigby, J. F. "A Concentrated Dose of Old-Fashioned Geometry." Math. Gaz. 57, 296-298, 1953.

Referenced on Wolfram|Alpha

Feuerbach Hyperbola

Cite this as:

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

Subject classifications