The Sum of the Trilinear Coordinates of a Point

Given a triangle , the trilinear coordinates of a point are the signed distances to the extended sides. Denote the signed distances of to , and by , and , respectively. If and the incenter are in the same half-plane determined by a side, the signed distance to that side is positive; otherwise, it is negative.

Let the triangle's circumcenter be , the circumradius be and the inradius be . Then the sum of the signed distances is:

.