Theorem 3: A quadrilateral is a parallelogram if and only if the diagonals bisect each other.
Since this is an "if and only if" proof, there are two things to prove.
1. Given: ABCD is a parallelogram
To prove: AE = EC, BE = ED
and the converse:
2. Given: AE = EC, BE = ED
To prove: ABCD is a parallelogram
There is another way to prove this.
The definition of a parallelogram is that the opposite sides are parallel. In the second way we establish that the opposite sides are parallel, so we can use the definition to conclude that the figure is a prarllelogram. It is simpler to show that the opposite sides are equal in length which we did in the first way. If we do it that way the reason that the figure is a parallelogram is that we proved that if the opposite sides are the same length then the figure is a parallelogram in Theorem 2.