KURSCHAK 1993 Problem 2

Triangle $ABC$ is not isosceles. The incircle of $\triangle ABC$ touches the sides $BC$ , $CA$ , $AB$ in the points $K$ , $L$ , $M$ . The parallel with $LM$ through $B$ meets $KL$ at $D$ , the parallel with $LM$ through $C$ meets $KM$ at $E$ .

Prove that $DE$ passes through the midpoint of $\overline{LM}$ .