NMC 2024 Problem 2

Let $\mathcal{Q}_{1}$ be a quadrilateral such that the midpoints of its sides lie on a circle. Prove that there exists a cyclic quadrilateral $\mathcal{Q}_{2}$ with the same sidelengths as $\mathcal{Q}_{1}$, such that two of the angles in $\mathcal{Q}_{2}$ are equal.