Trilinear Coordinates - Examples

Examples

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

where a, b, c are the respective sidelengths BC, CA, AB, and σ = 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.)

• A = 1 : 0 : 0
• B = 0 : 1 : 0
• C = 0 : 0 : 1
• incenter = 1 : 1 : 1
• centroid = bc : ca : ab = 1/a : 1/b : 1/c = csc A : csc B : csc C.
• circumcenter = cos A : cos B : cos C.
• orthocenter = sec A : sec B : sec C.
• nine-point center = cos(B − C) : cos(C − A) : cos(A − B).
• symmedian point = a : b : c = sin A : sin B : sin C.
• A-excenter = −1 : 1 : 1
• B-excenter = 1 : −1 : 1
• C-excenter = 1 : 1 : −1.

Note that, in general, the incenter is not the same as the centroid; the centroid has barycentric coordinates 1 : 1 : 1 (these being proportional to actual signed areas of the triangles BGC, CGA, AGB, where G = centroid.)