Last edited by Albert; December 28th, 2013 at 01:07.
Let $r$ be the ray which passes through $M$ and whose beginning point is $B$.
Let $P$ be a point such that $\angle PBA=\angle ABC$.
Draw a ray $l$ which is perpendicular to $AB$ and whose beginning point is $P$.
Let $l$ intersect $BC$ at $N'$.
Let Find a point $M'$ on $r$ such that $|N'M'|=|PN'|$.
Now find a point $N$ on $BC$ such that $\Delta M'BN'\sim MBN$. Then $N$ is the required point.