KöMaL 2013 A599

Two parabolas, $\mathcal{P}_1$ and $\mathcal{P}_2$ have the same focus point. The directrix of $\mathcal{P}_1$ meets $\mathcal{P}_2$ at points $A$ and $B$. The directrix of $\mathcal{P}_2$ meets $\mathcal{P}_1$ at $C$ and $D$. Show that the points $A$, $B$, $C$ and $D$ are concyclic.