  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.
