Large disks touching three sides of a quadrilateral

We show that every Jordan quadrilateral $Q\subset \mathbb{C}$ contains a disk $D$ so that $\mathrm{\partial }D\cap \mathrm{\partial }Q$ 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.