Given a line segment $\overline{AB}$ and a point $C$ on $\overline{AB}$ construct a line perpendicular to $\overline{AB}$ that is incident with $C$.
I know how to construct a normal perpendicular bisector but I'm not sure how to do it with a given point on the line. For our constructions we are only allowed a compass and straightedge, no computer software. Any suggestions?
Center the compass in $C$ to find two points $A'$, $B'\in AB$ such that $C$ is the midpoint of $A'B'$, and then find the perpendicular bisector to $A'B'$.