Bicentric configurations of pentagonal linkages

Let $P$ be a planar $n$-gon with the sidelengths $a_1,\dots ,a_n$ and let us denote by $L=L(P)$ the corresponding planar polygonal linkage. We are concerned with the problem of finding conditions on the sidelengths $a_i$ which guarantee the existence of a bicentric configuration of $L$. Specifically, we present some related results on tangential and chordal polygons and characterize pentagonal linkages having bicentric configurations.