Prove that the line segments joining the midpoints of opposite sides of a quadrilateral bisect each other.

In ΔADC, S and R are the midpoints of AD and DC respectively.

By midpoint theorem,
Hence SR || AC and SR = AC … (1)
Similarly, in ΔABC, P and Q are midpoints of AB and BC respectively.
PQ || AC and PQ = AC …(2) …By midpoint theorem
From equations (1) and (2), we get
PQ || SR and PQ = SR …(3)
Here, one pair of opposite sides of quadrilateral PQRS is equal and parallel.
Hence PQRS is a parallelogram
Hence the diagonals of parallelogram PQRS bisect each other.
Thus PR and QS bisect each other.

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