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?