Use equal vectors and diagonal lengths to identify a quadrilateral

Referred to an origin $O$, the position vectors of four non-collinear points $A$, $B$, $C$ and $D$ are $\mathbf a$, $\mathbf b$, $\mathbf c$ and $\mathbf d$ respectively. Given that $\mathbf a-\mathbf b=\mathbf d-\mathbf c$, show that $ABCD$ is a parallelogram.

Given also that $|\mathbf a-\mathbf c|=|\mathbf b-\mathbf d|$, identify the shape of the parallelogram $ABCD$, justify your answer.