Incircles of trilaterals

Abstract

A trilateral is a Jordan domain $G$ whose boundary is decomposed into three Jordan arcs $\alpha , \beta$ and $\gamma$. A circle $C$ which is contained in the closure of $G$ and touches $\alpha , \beta$ and $\gamma$ is called an incircle of the trilateral. Using Sperner’s Lemma, we prove that every trilateral has an incircle. If the boundary of $G$ is smooth (or, more generally, if the trilateral is “tame”), then its incircle is uniquely determined. These results are extended to general simply connected domains in the framework of prime ends.