Intriguing Invariants of Centers of Ellipse-Inscribed Triangles

TL;DR

This paper investigates geometric invariants related to centers of triangles inscribed in ellipses, revealing new properties about their loci and invariants under specific vertex configurations.

Contribution

It introduces novel invariants of ellipse-inscribed triangle centers and characterizes their loci and invariants under affine combinations and vertex configurations.

Findings

Locus of certain triangle centers is an ellipse.

Centers of affine combinations sweep a line.

Loci translate rigidly along a line.

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.