Intriguing Invariants of Centers of Ellipse-Inscribed Triangles
arXiv:2010.09408 · doi:10.1007/s00022-021-00586-3
Abstract
We describe invariants of centers of ellipse-inscribed triangle families with two vertices fixed to the ellipse boundary and a third one which sweeps it. We prove that: (i) if a triangle center is a fixed affine combination of barycenter and orthocenter, its locus is an ellipse; (ii) and that 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. Additionally, we study invariants of the envelope of elliptic loci over combinations of two fixed vertices on the ellipse.
17 pages, 10 figures, 3 tables, 6 video links