Given, a quadrilateral ABCD with AB = DC and DO = OB.
To prove: ABCD a parallelogram
Construction: Draw OX DC.
In Δ BCD,
O is the mid point of BD and OX DC.
∴By converse of mid point theorem,
X is the mid point of BC and OX = DC.
As AB = DC
∴ OX = AB.
Now, in Δ ABC,
X is the mid point of BC and OX = AB.
∴ By converse of mid point theorem,
O is the mid point of AC and OX AB
As OX AB and OX DC
⇒ AB DC
Now, AB DC and AB = DC
∴ ABCD is a parallelogram.