P and Q are points on opposite sides AD and BC of a parallelogram ABCD such that PQ passes through the point of intersection O of its diagonals AC and BD. Show that PQ is bisected at O.

If ABCD is a rectangle in which the diagonal BD bisects ∠B, then show that ABCD is a square.

In figure (1) given below, ABCD is a parallelogram and X is mid-point of BC. The line AX produced meets DC produced at Q. The parallelogram ABPQ is completed. Prove that

(i) the triangles ABX and QCX are congruent.

(ii) DC = CQ = QP

If the diagonals of a quadrilateral are equal and bisect each other at right angles, then prove that it is a square.