
Journal for Geometry and Graphics 26 (2022), No. 2, 253269 Copyright Heldermann Verlag 2022 From Permutation Points to Permutation Cubics Boris Odehnal University of Applied Arts, Vienna, Austria boris.odehnal@uniak.ac.at 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 γ 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 γ. In the present paper we shall discuss especially the cases where γ is the isogonal or the isotomic conjugation. Properties and remarkable features of these cubics shall be elaborated. Keywords: Permutation point, triangle cubic, permutation cubic, triangle center, antiorthic axis, Mandart circumellipse. MSC: 14H45; 51N35 [ Fulltextpdf (597 KB)] 