For a triangle , nine points lie on its nine-point circle : the midpoints , , ; the feet of the altitudes , , ; and the midpoints , , of the line segments joining the vertices to the orthocenter .

The nine-point center of is the center of its nine-point circle [1].

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

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

Let , set , , to be the Conway notation, and let be the foot of the perpendicular from to .

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 have a sum of 1.