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)
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(2)
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(3)
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The constant is given by
(4)
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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)
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(6)
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(7)
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(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 |