The Feuerbach triangle is the triangle formed by the three points of tangency of the nine-point circle with the excircles (Kimberling 1998, p. 158). (The fact that the excircles touch the nine-point circle is known as Feuerbach's theorem.)
The Feuerbach triangle has trilinear vertex matrix
The circumcenter of the Feuerbach triangle is the nine-point center of the reference triangle.
If and are the incenter and nine-point center of a triangle and is its Feuerbach point, then and its Feuerbach triangle are perspective, and the perspector is the harmonic conjugate of with respect to the segment . Equivalently, the perspector is the internal similitude center of the incircle and the nine-point circle.