The talented Mr. Inversive Triangle in the elliptic billiard

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 limacon. 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.

Automated summary

The study demonstrates that inverting the vertices of elliptic billiard N-periodics with respect to a circle centered on one focus yields a new focus-inversive family with surprising invariants. This family has interesting properties such as a stationary Gergonne point, being a 3-periodic family of a second elliptic billiard, and having various triangle centers with circular loci.