Let the angle bisectors of $A$, $B$ meet again at $U$ and let the other pair meet again at $V$. It is easy to see that $\angle AUB = \angle CVD= 90^\circ$, so $UVXY$ is cyclic. Since $\mathrm{dist}(U, AD) = \mathrm{dist}(U, AB) = \mathrm{dist}(U, BC)$ and similarly $\mathrm{dist}(V, AD) = \mathrm{dist}(V, BC)$, we have that $UV\parallel AD\parallel BC$. Now, the converse of Reim's theorem finishes.