Relating Trilinear and Tripolar Coordinates for a Triangle

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.

The tripolar coordinates of the point are its distances to the vertices of the triangle, given by , and .

The Conway triangle notation relates the sides to twice the area of the triangle, denoted by :

, , , .

These definitions imply the following formulas between the trilinear and tripolar coordinates:

, , .