ISL 2002 Problem 8

Let two circles $S_{1}$ and $S_{2}$ meet at the points $A$ and $B$ . A line through $A$ meets $S_{1}$ again at $C$ and $S_{2}$ again at $D$ . Let $M$ , $N$ , $K$ be three points on the line segments $CD$ , $BC$ , $BD$ respectively, with $MN$ parallel to $BD$ and $MK$ parallel to $BC$ . Let $E$ and $F$ be points on those arcs $BC$ of $S_{1}$ and $BD$ of $S_{2}$ respectively that do not contain $A$ . Given that $EN$ is perpendicular to $BC$ and $FK$ is perpendicular to $BD$ prove that $\angle EMF=90^{\circ}$ .

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