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Given a triangle ABC, the trilinear coordinates of a point P, with respect to triangle ABC, are an ordered triple of numbers, each of which is proportional to the directed distance from P to one of the side lines.
If point P has trilinear coordinates [alpha] : [beta] : [gamma], then the Cartesian coordinates of P are calculated as follows:
Let N have the trilinear coordinates [alpha] : [beta] : [gamma].
Using (9) and (12), we find trilinear coordinates [alpha] : [beta] : [gamma] of Nagel point N of triangle ABC as follows: