Suppose ABC is a triangle. According to the very famous Feuerbach Theorem, the incircle
and nine-point circle meet in a point. Labeled F in the figure, it is called the Feuerbach point.
Trilinear coordinates for F are
1 - cos(B - C):
The points labeled Fa, Fb, Fc are the vertices of the Feuerbach Triangle. Although not shown in the figure, the lines A-to-Fa, B-to-Fb, C-to-Fc meet in the harmonic conjugate of F, having trilinears
1 + cos(B - C):
A discussion of the Feuerbach point and biography of K. W. Feuerbach (1800-1834) are given in
Dan Pedoe, Circles: A Mathematical View, Mathematical Association of America, Washington, D. C., 1995.