Point where the incircle and nine-point circle of a triangle are tangent
Feuerbach's theorem: the nine-point circle is tangent to the incircle and excircles of a triangle. The incircle tangency is the Feuerbach point.
In the geometry of triangles, the incircle and nine-point circle of a triangle are internally tangent to each other at the Feuerbach point of the triangle. The Feuerbach point is a triangle center, meaning that its definition does not depend on the placement and scale of the triangle. It is listed as X(11) in Clark Kimberling's Encyclopedia of Triangle Centers, and is named after Karl Wilhelm Feuerbach.[1][2]
Feuerbach's theorem, published by Feuerbach in 1822,[3] states more generally that the nine-point circle is tangent to the three excircles of the triangle as well as its incircle.[4] A very short proof of this theorem based on Casey's theorem on the bitangents of four circles tangent to a fifth circle was published by John Casey in 1866;[5] Feuerbach's theorem has also been used as a test case for automated theorem proving.[6] The three points of tangency with the excircles form the Feuerbach triangle of the given triangle.
^Kimberling, Clark (1994), "Central Points and Central Lines in the Plane of a Triangle", Mathematics Magazine, 67 (3): 163–187, doi:10.1080/0025570X.1994.11996210, JSTOR 2690608, MR 1573021.
^Encyclopedia of Triangle Centers Archived April 19, 2012, at the Wayback Machine, accessed 2014-10-24.
^Feuerbach, Karl Wilhelm; Buzengeiger, Carl Heribert Ignatz (1822), Eigenschaften einiger merkwürdigen Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten Linien und Figuren. Eine analytisch-trigonometrische Abhandlung (Monograph ed.), Nürnberg: Wiessner.
^Scheer, Michael J. G. (2011), "A simple vector proof of Feuerbach's theorem" (PDF), Forum Geometricorum, 11: 205–210, arXiv:1107.1152, MR 2877268.
^Casey, J. (1866), "On the Equations and Properties: (1) of the System of Circles Touching Three Circles in a Plane; (2) of the System of Spheres Touching Four Spheres in Space; (3) of the System of Circles Touching Three Circles on a Sphere; (4) of the System of Conics Inscribed to a Conic, and Touching Three Inscribed Conics in a Plane", Proceedings of the Royal Irish Academy, 9: 396–423, JSTOR 20488927. See in particular the bottom of p. 411.
^Chou, Shang-Ching (1988), "An introduction to Wu's method for mechanical theorem proving in geometry", Journal of Automated Reasoning, 4 (3): 237–267, doi:10.1007/BF00244942, MR 0975146, S2CID 12368370.
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