Hence, we know that $\bfd - \bfa$ and $\bfc - \bfb$ are in the same direction and have equal length. We would like to show that $\bfb - \bfa$ and $\bfc - \bfd$ are in the same direction.

In order to relate the vectors, we need to use the information that $\bfa\bfd$ and $\bfb\bfc$.

Because $\bfd - \bfa$ and $\bfc-\bfb$ have the same length and direction, they are equal as vectors:$$\bfd - \bfa = \bfc - \bfb.$$

We are after a relation involving $\bfc - \bfd$, so we add $\bfb$ and subtract $\bfd$ from both sides: $$\bfb - \bfa = \bfc - \bfd.$$

Hence, $\bfa\bfb$ and $\bfd\bfc$ have equal length and direction.

We conclude that the quadrilateral is a parallelogram.