## Problem on geometric proofs with vectors

Suppose that two opposite sides of a quadrilateral are parallel and have equal length. Show that the quadrilateral is a parallelogram.
• ## Solution

Because we are trying to show that $\bfa\bfb\bfc\bfd$ is a parallelogram, we recall
A parallelogram is a quadrilateral that has two pairs of parallel sides.
Suppose that $\bfa\bfd$ and $\bfb\bfc$ are the parallel sides with equal length. We need to show that $\bfa\bfb$ and $\bfd\bfc$ are parallel.
We express each side as a vector.
Recall that
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.