Let ABCD be a quadrilateral. E, F, G and H are the midpoints of sides AB, BC, CD and DA respectively.

We need to prove EG and HF bisect each other. It is sufficient to show EFGH is a parallelogram, as the diagonals in a parallelogram bisect each other.

Let the position vectors of these vertices and midpoints be as shown in the figure.

As E is the midpoint of AB, using midpoint formula, we have

Similarly, , and .

Recall the vector is given by

Similarly

So, we have .

Two vectors are equal only when both their magnitudes and directions are equal.

and.

This means that the opposite sides in quadrilateral EFGH are parallel and equal, making EFGH a parallelogram.

EG and HF are diagonals of parallelogram EFGH. So, EG and HF bisect each other.

Thus, the line segments joining the midpoints of opposite sides of a quadrilateral bisect each other.