Consider the below quadrilateral ABCD in which diagonals bisect each other.
ABCD is a quadrilateral.
AC & BD are diagonals bisecting each other.
Hence, AO = OC and BO = OD.
Prove that the given quadrilateral is a parallelogram.
Consider triangle AOB and COD;
AO = OC (given)
∠AOB = ∠COD ( vertically opposite angle)
BO = OD (given)
By SAS congruency condition, both triangles AOB & COD are congruent.
Since both triangles are congruent, we can write;
∠BAO = ∠DCO
Both the angles are also alternate interior angles formed by line AB & CD intersected by transversal AC.
Since both the angles are equal this means that the line AB & CD are parallel to each other.
Hence, AB || CD.
Now consider the triangle AOD and COB.
OA = OC (given)
∠AOD = ∠COB ( vertically opposite angle)
OD = OB
By SAS congruency, both triangles AOD & COB are congruent.
Since both triangles are congruent, we can write ∠ODA = ∠OBC.
These are also alternate interior angles formed by line AD & BC intersected by transversal BD.
Since both angles are equal, it means the line AD & BC are parallel.
Hence, AD || BC.
Combining all the results we get;
AD || BC and AB || CD.
Since opposite sides are parallel to each other, the given quadrilateral is a parallelogram.
Hence in quadrilateral, if diagonal bisect each other it is a parallelogram.