PAMO 2015 Problem 6

Let $ABCD$ be a quadrilateral (with non-perpendicular diagonals).

The perpendicular from $A$ to $BC$ meets $CD$ at $K$ . The perpendicular from $A$ to $CD$ meets $BC$ at $L$ . The perpendicular from $C$ to $AB$ meets $AD$ at $M$ . The perpendicular from $C$ to $AD$ meets $AB$ at $N$ .

1. Prove that $KL$ is parallel to $MN$ .
2. Prove that $KLMN$ is a parallelogram if $ABCD$ is cyclic.