## Trilinear Coordinates

Given a 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 Barycentric Coordinates, Homogeneous Coordinates, or trilinears.''

In trilinear coordinates, the three Vertices , , and are given by , , and . 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

 (1)

so
 (2)

where is the Area of and , , and are the lengths of its sides. When the values of the coordinates are taken as the actual lengths (i.e., the trilinears are chosen so that ), the coordinates are known as Exact Trilinear Coordinates.

Trilinear coordinates are unchanged when each is multiplied by any constant , so

 (3)

When normalized so that
 (4)

trilinear coordinates are called Areal Coordinates. The trilinear coordinates of the line
 (5)

are
 (6)

where is the Point-Line Distance from Vertex to the Line.

Trilinear coordinates for some common Points are summarized in the following table, where , , and are the angles at the corresponding vertices and , , and are the opposite side lengths.

 Point Trilinear Center Function Centroid , Circumcenter de Longchamps Point Equal Detour Point Feuerbach Point Incenter 1 Isoperimetric Point Lemoine Point Nine-Point Center Orthocenter Vertex Vertex Vertex

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

 (7) (8)

where these are the Unit Vectors and . Assume the Triangle has been labeled such that is the lower rightmost Vertex and . Then the Vectors obtained by traveling and along the sides and then inward Perpendicular to them must meet

 (9)

Solving the two equations
 (10) (11)

gives

 (12) (13)

But and are Unit Vectors, so

 (14) (15)

And the Vector coordinates of the point are then
 (16)

References

Boyer, C. B. History of Analytic Geometry. New York: Yeshiva University, 1956.

Casey, J. The General Equation--Trilinear Co-Ordinates.'' Ch. 10 in A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions, with Numerous Examples, 2nd ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 333-348, 1893.

Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, pp. 67-71, 1959.

Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, 1969.

Coxeter, H. S. M. Some Applications of Trilinear Coordinates.'' Linear Algebra Appl. 226-228, 375-388, 1995.

Kimberling, C. Triangle Centers and Central Triangles.'' Congr. Numer. 129, 1-295, 1998.