Abstract
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.
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