Forum Geometricorum 7 (2007) 67--72.

Abstract: Suppose L_1 and L_2 are lines. There exists a unique point U such that if X in L_1, then X^{-1}© U in L_2, where X^{-1} denotes the isogonal conjugate of X and X^{-1}© U is the X^{-1}-Ceva conjugate of U. The mapping X -> X^{-1}© U is the U-Ceva collineation. It maps every line onto a line and in particular maps L_1 onto L_2. Examples are given involving the line at infinity, the Euler line, and the Brocard axis. Collineations map cubics to cubics, and images of selected cubics under certain U-Ceva collineations are briefly considered.

[ps file] [pdf]