Given a triangle , Feuerbach's theorem states that the nine-point circle is tangent to the three excircles and the incircle of . The point of tangency of with the incircle is called the Feuerbach point, [1].

Let , , be the exact trilinear coordinates of with respect to and let .

Let , , be the side lengths opposite the corresponding vertices and let , , be the circumradius, inradius and semiperimeter of .

A triangle center is said to be even when its barycentric coordinates can be expressed as a function of three variables , , that all occur with even exponents. If the center of a triangle has constant barycentric coordinates, it is called a neutral center (the centroid is the only neutral center). A triangle center is said to be odd if it is neither even nor neutral.

Standard barycentric coordinates of a point with respect to a reference triangle are normalized to a sum of 1.