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
Begin a geometric proof by labeling important points

We begin by labeling the four vertices of the quadrilateral as vectors, $\mathbf{o}, \mathbf{a}, \mathbf{b}, \mathbf{c}$.
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
Midpoint between two vectors

The midpoint between the vectors $\mathbf{x}$ and $\mathbf{y}$ is $\frac{\mathbf{x} + \mathbf{y}}{2}$.
We label all midpoints of the edges of the quadrilateral:
We now write the two connecting line segments as vectors. Recall that
Subtraction gives the vector between two points

The vector from $\mathbf{x}$ to $\mathbf{y}$ is given by $\mathbf{y} - \mathbf{x}$.
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.

Problem on geometric proofs with vectors

Solution

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