August Möbius introduced the system of barycentric or areal coordinates in 1827 [1, 2]. The idea is that one may attach weights to points, and that a system of weights determines a centre of mass. Given a triangle ABC, one obtains a coordinate system for the plane by placing weights x, y and z at the vertices (with x + y + z = 1) to describe the point which is the centre of mass. The vertices have coordinates A = (1, 0, 0), B = (0, 1, 0) and C = (0, 0, 1). By scaling so that triangle ABC has area [ABC] = 1, we can take the coordinates of P to be ([PBC], [PCA], [PBA]), provided that we take area to be signed. Our convention is that anticlockwise triangles have positive area.
Points inside the triangle have strictly positive coordinates, and points outside the triangle must have at least one negative coordinate (we are allowed negative masses). The equation of a line looks like the equation of a plane in Cartesian coordinates, but note that the equation lx + ly + lz = 0 (with l ≠ 0) is not satisfied by any point (x, y, z) of the Euclidean plane. We use such equations to describe the line at infinity when doing projective geometry.