Exact Trilinear Coordinates

The trilinear coordinates alpha:beta:gamma of a point P relative to a reference triangle are proportional to the directed distances a^':b^':c^' from P to the side lines of the triangle, but are undetermined up to a constant of proportionality k, i.e.,


The constant k is given by


where Delta=rs is the triangle area of DeltaABC, r is the inradius, s is the semiperimeter, and a, b, and c are the lengths of its sides.

The directed distances a^', b^', c^' themselves are called "exact" (or "actual") trilinear coordinates, and denoted (a^',b^',c^'). Therefore, if the trilinears alpha:beta:gamma are given for a point P, then its exact trilinears (a^',b^',c^') can be calculated according to


(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 R is the circumradius and r is the inradius.

See also

Areal Coordinates, Barycentric Coordinates, Reference Triangle, Trilinear Coordinates

Explore with Wolfram|Alpha


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.

Referenced on Wolfram|Alpha

Exact Trilinear Coordinates

Cite this as:

Weisstein, Eric W. "Exact Trilinear Coordinates." From MathWorld--A Wolfram Web Resource.

Subject classifications