MEMO 2012 Problem 5

Let $K$ be the midpoint of the side $AB$ of a given triangle $ABC$ . Let $L$ and $M$ be points on the sides $AC$ and $BC$ , respectively, such that $\angle CLK = \angle KMC$ . Prove that the perpendiculars to the sides $AB, AC,$ and $BC$ passing through $K,L,$ and $M$ , respectively, are concurrent.