OSN 2005 Problem 7

Let $ABCD$ be a convex quadrilateral. Square $AB_1A_2B$ is constructed such that the two vertices $A_2,B_1$ is located outside $ABCD$ . Similarly, we construct squares $BC_1B_2C$ , $CD_1C_2D$ , $DA_1D_2A$ . Let $K$ be the intersection of $AA_2$ and $BB_1$ , $L$ be the intersection of $BB_2$ and $CC_1$ , $M$ be the intersection of $CC_2$ and $DD_1$ , and $N$ be the intersection of $DD_2$ and $AA_1$ . Prove that $KM$ is perpendicular to $LN$ .