High Quality Content by WIKIPEDIA articles! In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative distances from the three sides of the triangle. Trilinear coordinates are an example of homogeneous coordinates. They are often called simply "trilinears". The incenter has trilinears 1 : 1 : 1; that is, the (directed) distances from the incenter to the sidelines BC, CA, AB of a triangle ABC are proportional to the actual distances, which are the ordered triple (r, r, r), where r is the inradius of triangle ABC. Note that the notation x:y:z using colons distinguishes trilinears from actual directed distances, (kx, ky, kz), which is the usual notation for an ordered triple, and which may be obtained from x : y : z using the number k = frac{2sigma}{ax + by + cz} where a, b, c are the respective sidelengths BC, CA, AB, and = area of ABC. ("Comma notation" for trilinears should be avoided, because the notation (x, y, z), which means an ordered triple, does not allow, for example, (x, y, z) = (2x, 2y, 2z), whereas the "colon notation" does allow x : y : z = 2x : 2y : 2z.)

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