Feuerbach's Conic Theorem

The locus of the centers of all circumconics that also pass through the orthocenter of a triangle (which, when not degenerate, are rectangular hyperbolas) is a circle through the midpoints of the sides, the points half way from the orthocenter to the vertices, and the feet of the altitude (Coolidge 1959, p. 198; Eddy and Fritsch 1994). Moreover, this circle is the nine-point circle (Kimberling 1998, p. 236).

This theorem is attributed to Feuerbach by Coolidge (1959), but does not appear in Feuerbach's treatise (Feuerbach 1822; Eddy and Fritsch 1994). In fact, it first appeared in Brianchon and Poncelet (1821).

See also

Altitude, Circumconic, Circumhyperbola, Conic Section, Feuerbach Hyperbola, Feuerbach's Theorem, Kiepert Hyperbola, Jerabek Hyperbola, Midpoint, Nine-Point Circle, Orthocenter, Rectangular Hyperbola

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Brianchon, C.-J. and Poncelet, J.-V. "Recherches sur la détermination d'une hyperbole équilatère, au moyen de quatre conditions données." Ann. des Math. 11, 205-220, 1821.Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, 1959.Eddy, R. H. and Fritsch, R. "The Conics of Ludwig Kiepert: A Comprehensive Lesson in the Geometry of the Triangle." Math. Mag. 67, 188-205, 1994.Feuerbach, K. W. Eigenschaften einiger merkwürdigen Punkte des geradlinigen Dreiecks, und mehrerer durch die bestimmten Linien und Figuren. Nürnberg, Germany: Riegel und Wiesner, 1822.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.

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Feuerbach's Conic Theorem

Cite this as:

Weisstein, Eric W. "Feuerbach's Conic Theorem." From MathWorld--A Wolfram Web Resource.

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