Trilinear Coordinates

Given a reference triangle , the trilinear coordinates of a point with respect to are an ordered triple of numbers, each of which is proportional to the directed distance from to one of the side lines. Trilinear coordinates are denoted or and also are known as homogeneous coordinates or "trilinears." Trilinear coordinates were introduced by Plücker in 1835. Since it is only the ratio of distances that is significant, the triplet of trilinear coordinates obtained by multiplying a given triplet by any nonzero constant describes the same point, so

(1)

For simplicity, the three polygon vertices , , and of a triangle are commonly written as , , and , respectively.

Trilinear coordinates can be normalized so that they give the actual directed distances from to each of the sides. To perform the normalization, let the point in the above diagram have trilinear coordinates and lie at distances , , and from the sides , , and , respectively. Then the distances , , and can be found by writing for the area of , and similarly for and . We then have

			(2)
			(3)
			(4)
			(5)

so

(6)

where is the area of and , , and are the lengths of its sides (Kimberling 1998, pp. 26-27). To obtain trilinear coordinates giving the actual distances, take , so we have the coordinates

(7)

These normalized trilinear coordinates are known as exact trilinear coordinates.

The trilinear coordinates of the line

(8)

are

(9)

where is the point-line distance from polygon vertex to the line.

The homogeneous barycentric coordinates corresponding to trilinear coordinates are , and the trilinear coordinates corresponding to homogeneous barycentric coordinates are .

Important points of a triangle are called triangle centers, and the vector functions describing the location of the points in terms of side length, angles, or both, are called triangle center functions . Since by symmetry, triangle center functions are of the form

(10)

it is common to call the scalar function "the" triangle center function. Note also that side lengths and angles are interconvertible through the law of cosines, so a triangle center function may be given in terms of side lengths, angles, or both. Trilinear coordinates for some common triangle centers are summarized in the following table, where , , and are the angles at the corresponding vertices and , , and are the opposite side lengths. Here, the normalizations have been chosen to give a simple form.

triangle center	triangle center function
circumcenter
de Longchamps point
equal detour point
Feuerbach point
incenter	1
isoperimetric point
symmedian point
nine-point center
orthocenter
triangle centroid	,

In trilinear coordinates, the coordinates of the vertices are 1:0:0 (), 0:1:0 (), and 0:0:1 (). Extensions along the sidelines by a distance have trilinears as illustrated above.

Trilinear coordinates of points fractional distances , , and along the sidelines are given in the above figure, where .

A point located a fraction of the distance along the sideline from to has trilinear coordinates

(11)

To determine the conversion of trilinear to Cartesian coordinates, orient the triangle with the axis parallel to the -axis and with its incenter at the origin, as illustrated above. Then

			(12)
			(13)

where

(14)

is the inradius, is the triangle area, and

(15)

(Kimberling 1998, pp. 31-33).

More generally, to convert trilinear coordinates to a vector position for a given triangle specified by the - and -coordinates of its axes, pick two unit vectors along the sides. For instance, pick

			(16)
			(17)

where these are the unit vectors and . Assume the triangle has been labeled such that is the upper rightmost polygon vertex and . Then the vectors obtained by traveling and along the sides and then inward perpendicular to them must meet

(18)

Solving the two equations

			(19)
			(20)

gives

			(21)
			(22)

But and are unit vectors, so

			(23)
			(24)

And the vector coordinates of the point are then

(25)