Show that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other.

Let ABCD be a quadrilateral and P, Q, R, and S the midpoints of AB, BC, CD, and DA, respectively.

In ΔACD,

R and S are the midpoints of CD and DA, respectively.

SR || AC.

Similarly, 

PQ || AC

PS || BD

QR || BD

PQRS is parallelogram.

PR and QS are the diagonals of the parallelogram PQRS.

So, they will bisect each other.