I need help with this hard geometry problem.
Consider a trapezoid ABCD (AB||CD and AD=BC). A square $A_1A_2A_3A_4$ is inscribed in the trapezoid such that $A_1 \in AB$, $A_2\in BC$, $A_3\in CD$ and $A_4 \in AD$. If $M$ and $N$ are the midpoints of AD and BC and $A_1A_3 \cap A_2A_4 = O$, prove that $O \in MN$.