We know that a quadrilateral is a parallelogram if the co-ordinates of mid-point s of its both the diagonals are same.

Therefore, we’ll find the mid-point s of diagonal AC and BD.

Let the co-ordinates of mid-point of AC be (x3, y3).

And, we know mid-point formula, i.e. the coordinates of mid-point of line joining (x1, y1) and (x2, y2) is

(Where, (x1, y1) and (x2, y2) are the coordinates of A and C

⇒ Co – ordinates of mid-point of AC =

Now, Let the co-ordinates of mid-point of BD be (x4, y4).

And, since it is a mid-point –

(Where, (x5, y5) and (x6, y6) are the coordinates of B and D.

⇒ Co – ordinates of mid-point of BD =

and since these are equal

⇒ abcd is a parallelogram