IrMO 2015 Problem 4

Two circles $C_1$ and $C_2$ , with centres at $D$ and $E$ respectively, touch at $B$ . The circle having $DE$ as diameter intersects the circle $C_1$ at $H$ and the circle $C_2$ at $K$ . The points $H$ and $K$ both lie on the same side of the line $DE$ . $HK$ extended in both directions meets the circle $C_1$ at $L$ and meets the circle $C_2$ at $M$ . Prove that (a) $|LH| = |KM|$ (b) the line through $B$ perpendicular to $DE$ bisects $HK$ .