The talented Mr. Inversive Triangle in the elliptic billiard

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 limaçon. 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.