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# Feuerbach's Theorem

There are two theorems commonly known as Feuerbach's theorem. The first states that circle which passes through the feet of the perpendiculars dropped from the polygon vertices of any triangle on the sides opposite them passes also through the midpoints of these sides as well as through the midpoint of the segments which join the polygon vertices to the point of intersection of the perpendicular. Such a circle is called a nine-point circle.

The proposition most frequently called Feuerbach's theorem states that the nine-point circle of any triangle is tangent internally to the incircle and tangent externally to the three excircles. This theorem was first published by Feuerbach (1822). Many proofs have been given (Elder 1960), with the simplest being the one presented by M'Clelland (1891, p. 225) and Lachlan (1893, p. 74).

Excircles, Feuerbach Point, Feuerbach Triangle, Hart Circle, Incircle, Midpoint, Nine-Point Circle, Perpendicular, Tangent

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## References

Altshiller-Court, N. College Geometry: A Second Course in Plane Geometry for Colleges and Normal Schools, 2nd ed., rev. enl. New York: Barnes and Noble, pp. 107, 273, and 290, 1952.Baker, H. F. Appendix to Ch. 12 in An Introduction to Plane Geometry. Cambridge, England: Cambridge University Press, 1943.Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, p. 39, 1971.Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 117-119, 1967.Dixon, R. Mathographics. New York: Dover, p. 59, 1991.Durell, C. V. Modern Geometry: The Straight Line and Circle. London: Macmillan, p. 117, 1928.Elder, A. E. "Feuerbach's Theorem: A New Proof." Amer. Math. Monthly 67, 905-906, 1960.F. Gabriel-Marie. Exercices de géométrie. Tours, France: Maison Mame, pp. 595-597, 1912.Feuerbach, K. W. Eigenschaften einiger merkwürdigen Punkte des geradlinigen Dreiecks, und mehrerer durch die bestimmten Linien und Figuren. Nürnberg, Germany: Riegel und Wiesner, 1822.Gallatly, W. "Feuerbach's Theorem." §63 in The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, p. 41, 1913.Kroll, W. "Elementarer Beweis des Satzes von Feuerbach." Praxis der Math. 40, 251-254, 1998.Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillan, 1893.M'Clelland, W. J. A Treatise on the Geometry of the Circle and Some Extensions to Conic Sections by the Method of Reciprocation, with Numerous Examples. London: Macmillan, 1891.Rouché, E. and de Comberousse, C. Traité de géométrie plane. Paris: Gauthier-Villars, pp. 307-309, 1900.Sawayama, Y. "Démonstration élémentaire du théorème de Feuerbach." L'enseign. math. 7, 479-482, 1905.Sawayama, Y. "8 nouvelles démonstrations d'un théorème relatif au cercle des 9 points." L'enseign. math. 13, 31-49, 1911.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. Middlesex, England: Penguin Books, pp. 76-77, 1991.

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Feuerbach's Theorem

## Cite this as:

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