Intriguing invariants of centers of ellipse-inscribed triangles

We describe invariants of centers ellipse-inscribed triangle families with two vertices fixed to the ellipse boundary and a third one which sweeps it. prove that: (i) if center is affine combination barycenter orthocenter, its locus an ellipse; (ii) that over family said combinations, loci sweep line; (iii) parallel vertices, rigidly translate along second line. Additionally, we study envelope elliptic combinations on ellipse.