The Talented Mr. Inversive Triangle in the Elliptic Billiard

TL;DR

This paper explores a novel focus-inversive family of triangles inscribed in a Pascal's Limaçon derived from elliptic billiards, revealing invariants and geometric properties including fixed centers and a stationary Gergonne point.

Contribution

It introduces a new focus-inversive family of triangles with invariants and geometric properties, extending elliptic billiard theory and triangle center loci analysis.

Findings

Perimeter, sum of cosines, and sum of distances are invariant.

The family has a stationary Gergonne point.

Loci of various triangle centers are circles.

Abstract

Inverting the vertices of elliptic billiard N-periodics with respect to a circle centered on one focus yields a new "focus-inversive" family inscribed in Pascal's Lima\c{c}on. The following are some of its surprising invariants: (i) perimeter, (ii) sum of cosines, and (iii) sum of distances from inversion center (the focus) to vertices. We prove these for the N=3 case, showing that this family (a) has a stationary Gergonne point, (b) is a 3-periodic family of a second, rigidly moving elliptic billiard, and (c) the loci of incenter, barycenter, circumcenter, orthocenter, nine-point center, and a great many other triangle centers are circles.