Question

Prove logically the diagonals of a parallelogram bisect each other. Show conversely that a quadrilateral in which diagonals bisect each other is a parallelogram.

Answer 1

In ΔAOD and ΔBOC:

Thus, from ASA congruency, we get:

∴AO = OC and BO = BD

Hence, the diagonals of the parallelogram bisect each other.

Now, conversely, let AO = OC and BO = OD.

In ΔAOD and ΔBOC, we have:

AO = OC (Given)

BO = OD (Given)

Thus, we get AD || BC.

Similarly, it can be shown that AB || CD.

Hence, a quadrilateral, in which the diagonals bisect each other, is a parallelogram.