The Feuerbach triangle is the triangle formed by the three points of tangency of the nine-point circle with the excircles. (The fact that the excircles touch the nine-point circle is known as Feuerbach's theorem.) The Feuerbach triangle has trilinear vertex matrix [-sin^2[1/2(B - C)] | cos^2[1/2(C - A)] | cos^2[1/2(A - B)] cos^2[1/2(B - C)] | -sin^2[1/2(C - A)] | cos^2[1/2(A - B)] cos^2[1/2(B - C)] | cos^2[1/2(C - A)] | -sin^2[1/2(A - B)]]. The circumcenter of the Feuerbach triangle is the nine-point center of the reference triangle.

Feuerbach point | Feuerbach's theorem | nine-point circle