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

Let ABCD is a quadrilateral in which P, Q, R, and S are the mid-points of sides AB, BC, CD, and DA respectively. Join PQ, QR, RS, SP, and BD

In ΔABD, S and P are the mid-points of AD and AB respectively. Therefore, by using mid-point theorem:

SP || BD and SP = BD (1)

Similarly in ΔBCD,

QR || BD and QR = BD (2)

From equations (1) and (2), we obtain

SP || QR and SP = QR

In quadrilateral SPQR, one pair of opposite sides is equal and parallel to each other

Therefore, SPQR is a parallelogram.

We know that diagonals of a parallelogram bisect each other

Hence, PR and QS bisect each other