Exact Trilinear Coordinates

The trilinear coordinates of a point relative to a reference triangle are proportional to the directed distances from to the side lines of the triangle, but are undetermined up to a constant of proportionality , i.e.,

			(1)
			(2)
			(3)

The constant is given by

(4)

where is the triangle area of , is the inradius, is the semiperimeter, and , , and are the lengths of its sides.

The directed distances , , themselves are called "exact" (or "actual") trilinear coordinates, and denoted . Therefore, if the trilinears are given for a point , then its exact trilinears can be calculated according to

			(5)
			(6)
			(7)

(Sommerville 1961, p. 157; Eddy and Fritsch 1994; Kimberling 1998, p. 28). Note that points on the line at infinity do not have exact trilinear coordinates.

Exact trilinears for a number of triangle centers are summarized in the table below, where is the circumradius and is the inradius.

triangle center	exact trilinear coordinates
circumcenter	, ,
incenter	,
nine-point center
orthocenter	,
Spieker center	,
symmedian point
triangle centroid