## Problem on geometric proofs with vectors

Consider an arbitrary quadrilateral. The two blue line segments connect the midpoints of adjacent sides. Using only vector addition and multiplication by constants, show that these line segments are parallel and have the same length.
• ## Solution

Because we have information about the line segments connecting midpoints, we find all the midpoints in terms of $\mathbf{o}$, $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$.
Recall that
We label all midpoints of the edges of the quadrilateral:
We now write the two connecting line segments as vectors. Recall that
The upper-left line segment is thus $$\frac{\mathbf{b} + \mathbf{c}}{2} - \frac{\mathbf{c} + \mathbf{o}}{2} = \frac{\mathbf{b} - \mathbf{o}}{2}.$$
Similarly, the lower-right line segment is $$\frac{\mathbf{a} + \mathbf{b}}{2} - \frac{\mathbf{a} + \mathbf{o}}{2} = \frac{\mathbf{b} - \mathbf{o}}{2}.$$
As vectors, these line segments are equal. Hence they are parallel and have equal length.