M and N are the points of trisection of the diagonal BD of a parallelogram ABCD. Prove that CN is parallel to AM .

Given: ABCD is a parallelogram, with BD as its diagonal. M and N are points on BD such that BM = MN = ND.

To proof: CN is parallel to AM.

Construction: Join diagonal AC.

Proof: We know that diagonals of a parallelogram bisects each other.

⇒ OA = OC and OD = OB

Subtracting BM from both sides,

⇒ OB - BM = OD - BM

⇒ OB - BM = OD - DN

⇒ OM = ON

Similarly, because OA = OC and the diagonals bisect each other, we see that AM and CN are parallel and equal in length.

The opposite sides AM and CN, as well as AN and CM, are parallel and equal. Hence, AMCN forms a parallelogram.

Hence, CN is parallel to AM.