It is more than just a quadrilateral. In fact it is going to be hard to pick.
These facts suit a square, a rectangle, a rhombus, and a parallelogram. And the above statement is true, but maybe a little harder to prove than the converse of the statement, which is the usual one you find.
The converse is "If you have a parallelagram, the diagonals bisect each other."
You might think a trapezoid deserves some mention. The diagonals of a trapezoid do not bisect each other.