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Prove that this figure is a rhombus

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

*Given that ***AC** is on the perpendicular bisector of **BD**, and that **BD** lies on the perpendicular bisector of **AC**. Prove that quadrilateral **ABCD** is 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?