The points that lie on the bisector of the segment $AB$ are on equal distance of its end points $A$ and $B$. Hence the desired point $C$ is the intersection of the bisector of $AB$ and the line $p$. The problem has a unique solution if $AB$ and $p$ are not perpendicular, if $AB$ and $p$ are perpendicular and $p$ is not the bisector of $AB$ then the problem doesn't have a solution and if $AB$ and $p$ are perpendicular and $p$ is the bisector of $AB$ then the problem has infinitely many solution because every point of $p$ is a solution.