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
![[-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)]].](/images/equations/FeuerbachTriangle/NumberedEquation1.svg) |
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.
