I am working through Serge Lang's book on geometry, and I am having trouble for some reason on this proof:

Given thatACis on the perpendicular bisector ofBD, and thatBDlies on the perpendicular bisector ofAC. Prove that quadrilateralABCDis a rhombus (a rhombus is a parallelogram whose four sides have the same length).

It is easy to prove that the four sides are of equal length, but I don't know how to prove that the pairs of sides are parallel. I think I should prove that the interior angles are supplementary, but I'm not sure how to do that. Any tips?