KöMaL 2012 B4458

Lines $a$ and $b$ are given lines in the plane, $A$ is a given point not lying on $a$, and $B$ is a given point not lying on $b$. If point $O$ does not coincide with $A$ or $B$, then there exists a transformation composed of a rotation about $O$and a central enlargement from $O$, such that $A$ is mapped to $B$. What is the locus of those points in the plane for which this transformation maps the intersection of lines $OA$ and $a$ to the intersection of lines $OB$ and $b$?