Loci and Envelopes of Ellipse-Inscribed Triangles

We study a family of ellipse-inscribed triangles with two vertices V1,V2 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 V1V2, said elliptic loci are rigidly-translating ellipses. Additionally, we study the external envelope of elliptic loci for fixed V1 and over all V2 on the ellipse. We show that (iv) the area of said envelope is invariant with respect to V1, and that (v) for the barycenter (resp. orthocenter), the envelope is an ellipse (resp. an affine image of Pascal's Lima\c{c}on).