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.