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.
^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, JSTOR20488927. 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, MR0975146, S2CID12368370.