TOPICS

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).

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

Explore with Wolfram|Alpha

More things to try:

References

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.

Referenced on Wolfram|Alpha

Feuerbach's Conic Theorem

Cite this as:

Weisstein, Eric W. "Feuerbach's Conic Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FeuerbachsConicTheorem.html