Let us consider a Cartesian plane having a parallelogram OABC in which O is the origin.

We have to prove that middle point of the opposite sides of a quadrilateral and the join of the mid-points of its diagonals meet in a point and bisect each other.

Let coordinates be A(0, 0).

So other coordinates will be B(x_{1 +} x₂, y₁), C(x₂, 0) ... refer figure.

Let P, Q, R and S be the mid-points of the sides AB, BC, CD, DA respectively.

By midpoint formula,

x = , y =

For midpoint P on AB,

x =, y =

∴ x = , y =

∴ Coordinate of P is ( , )

For midpoint Q on BC,

x =, y =

∴ x = , y =

∴ Coordinate of Q is ( , )

For R, we can observe that, R lies on x axis.

∴ Coordinate of R is ( , )

For midpoint S on OA,

x =, y =

∴ x = , y =

∴ Coordinate of S is ( , )

For midpoint of PR,

x = , y =

∴ x = , y =

∴ Midpoint of PR is ( , )

Similarly midpoint of QS is ( , )

Also, similarly midpoint of AC and OA is ( , )

Hence, midpoints of PR, QS, AC and OA coincide

∴We say that middle point of the opposite sides of a quadrilateral and the join of the mid-points of its diagonals meet in a point and bisect each other.