Bicentric configurations of pentagonal linkages

TL;DR

This paper investigates conditions under which pentagonal linkages can form bicentric configurations, focusing on geometric properties and characterizations related to tangential and chordal polygons.

Contribution

It provides new criteria and characterizations for bicentric configurations in pentagonal linkages, expanding understanding of geometric linkage configurations.

Findings

Derived conditions for bicentric configurations in pentagons

Characterized pentagonal linkages with bicentric configurations

Connected bicentric configurations to tangential and chordal polygons

Abstract

Let $P$ be a planar $n$-gon with the sidelengths $a_1, \ldots, a_n$ 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.