Intriguing invariants of centers of ellipse-inscribed triangles (Q2046173)

In this paper, the authors describe intriguing properties of a 1d family of triangles: two vertices are pinned to the boundary of an ellipse while a third one sweeps it. They prove that: (i) if a triangle center is a fixed affine combination of barycenter and orthocenter, its locus is an ellipse; (ii) over the family of said affine combinations, the centers of said loci sweep a line; (iii) over the family of parallel fixed vertices, said loci rigidly translate along a second line.