Statement of the Theorem: Prove that the lines joining the middle points of the adjacent sides of a quadrilateral form a parallelogram.

Proof: Let ABCD be a quadrilateral and length of its side AB is 2a.

Let us choose origin of rectangular cartesian co-ordinates at the vertex A and x-axis along the side AB and AY as the y-axis. Then, the co-ordinates of A and B are (0, 0) and (2a, 0) respectively. Referred to the chosen axes, let (2b, 2c) and (2d, 2e) be the co-ordinates of the vertices C and D respectively. If J, K, L, M be the mid-points of the sides AB, BC, CD, and, DA, respectively, then the co-ordinates of J, K, L and M are (a, 0 ), (a + b, c), (b + d, c + e) and (d, e) respectively.

Now, the co-ordinates of the mid-point of the diagonal JL of the quadrilateral JKLM are {(a + b + d)/2, (c + e)/2}

Again, the co-ordinates of the mid-point of the diagonal MK of the same quadrilateral are {(a + b + d)/2, (c + e)/2}.

Clearly, the diagonals JL and MK of the quadrilateral JKLM bisect each other at ((a + b + d)/2, (c + e)/2). Hence, the quadrilateral JKLM is a parallelogram. Proved.

Co-ordinate Geometry