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Journal for Geometry and Graphics 26 (2022), No. 2, 253--269
Copyright Heldermann Verlag 2022

From Permutation Points to Permutation Cubics

Boris Odehnal
University of Applied Arts, Vienna, Austria

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

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