From Permutation Points to Permutation Cubics*

The trilinear coordinates of a point V in the plane of a triangle can be permuted in six ways which yields the six permutation points of V. These six points always lie on a single conic, called the permutation conic. A natural variant or generalization seems to be: The six permutation points of V together with the six permutation points of V 's image under a certain quadratic Cremona transformation. comprise a set of twelve points that always lie on a single cubic which we shall call the permutation cubic of V with respect to.. In the present paper we shall discuss especially the cases where. is the isogonal or the isotomic conjugation. Properties and remarkable features of these cubics shall be elaborated.