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 ellipse, 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).

翻译：Poncelet多边形的1d家族在两个圆圈间相互交织,称为“双心家庭”。使用椭圆函数和Liouville的理论,我们显示(一)这个家庭有内在角共和的内角共和,和(二)与家庭限制点有关的踏板多边形有变化性周边。有趣的是,(一)和(二)两者也是椭圆面的圆形N周期性。此外,由于(二)中的双翼多边形与(二)中外向型圆形N周期与以聚焦为中心的椭圆形的反正相同,因此一个重要的必然结果是:(三) 外向型圆面圆形圆形圆形的焦点-反向型N-形圆形有恒定的周边。有趣的是,这些圆锥形也保留了它们的共性(N=4案例除外)。