Cubics defined from symmetric functions

Suppose a,b,c are algebraic indeterminates. Let X = x:y:z and U = u:v:w be homogeneous trilinear coordinates for points in the transfigured plane of a triangle ABC; that is, x,y,z,u,v,w are functions of a,b,c (which need not be sidelengths of a euclidean triangle). Cubic equations of the form f(x,y,z) = f(u,v,w), where f is of degree 3 and symmetric or antisymmetric in a,b,c, are discussed, typified by f(x,y,z) = (y+z)(z+x)(x+y)/(xyz). Extensions are made to the case that the coordinates for X and U are general homogeneous, with results stated in terms of trilinear coordinates.