 TOPICS # 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 Areal Coordinates, Barycentric Coordinates, Reference Triangle, Trilinear Coordinates

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## References

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. https://mathworld.wolfram.com/ExactTrilinearCoordinates.html