Cyclic quadrilaterals and smooth Jordan curves

For every smooth Jordan curve \(\gamma\) and cyclic quadrilateral \(Q\) in the Euclidean plane, we show that there exists an orientation-preserving similarity taking the vertices of \(Q\) to \(\gamma\). The proof relies on the theorem of Polterovich and Viterbo that an embedded Lagrangian torus in \(\mathbb{C}^2\) has minimum Maslov number 2.