Prove perpendicular bisectors of non-parallel lines intersect

Question

Suppose that $A$, $B$ and $C$ are points and that $AB$ and $BC$ are not parallel. Show that the perpendicular bisector of $AB$, $l$, and the perpendicular bisector of $BC$, $l'$, are not parallel and so intersect.

Answer 1

Assume otherwise. Then $AB$ is perpendicular to both $l$ and to its parallel $l'$. As $BC$ is also perpendicular to $l'$, we conclude that $AB$ and $BC$ are parallel.