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.,

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.

Eddy, R. H. and Fritsch, R. "The Conics of Ludwig Kiepert: A Comprehensive Lesson in the Geometry of the Triangle." Math. Mag.67,
188-205, 1994.Kimberling, C. "Triangle Centers and Central Triangles."
Congr. Numer.129, 1-295, 1998.Sommerville, D. M. Y.
Analytical
Conics, 3rd ed. London: G. Bell and Sons, 1961.