  The 1d family of Poncelet polygons interscribed between two circles is known
as the Bicentric family. Using elliptic functions and Liouville's theorem, we
show (i) that this family has invariant sum of internal angle cosines and (ii)
that the pedal polygons with respect to the family's limiting points have
invariant perimeter. Interestingly, both (i) and (ii) are also properties of
elliptic billiard N-periodics. Furthermore, since the pedal polygons in (ii)
are identical to inversions of elliptic billiard N-periodics with respect to a
focus-centered circle, an important corollary is that (iii) elliptic billiard
focus-inversive N-gons have constant perimeter. Interestingly, these also
conserve their sum of cosines (except for the N=4 case).
