Large disks touching three sides of a quadrilateral

Abstract

We show that every Jordan quadrilateral \(Q\subset\mathbb{C}\) contains a disk \(D\) so that \(\partial D\cap\partial Q\) contains points of three different sides of \(Q\). As a consequence, together with some modulus estimates from Lehto and Virtanen, we offer a short proof of the main result obtained by Chrontsios-Garitsis and Hinkkanen in 2024 and it also improves the bounds on their result.