Intriguing Invariants of Centers of Ellipse-Inscribed Triangles

Abstract

We study a family of ellipse-inscribed triangles with two vertices V 1 , V 2 fixed on the ellipse boundary while a third one which sweeps it. We prove that: (i) if a triangle center is a fixed linear combination of barycenter and orthocenter, its locus over the family is an ellipse; (ii) over the 1d family of said linear combinations, loci centers sweep a line; (iii) over the family of parallel V 1 V 2 , said elliptic loci are rigidly-translating ellipses. Additionally, we study the external envelope of elliptic loci for fixed V 1 and over all V 2 on the ellipse. We show that (iv) the area of said envelope is invariant with respect to V 1 , and that (v) for the barycenter (resp. orthocenter), the envelope is an ellipse (resp. an affine image of Pascal's Limaçon).