Question
Show that the line joining one vertex of a parallelogram to the mid-point of an opposite side trisects the diagonal and is trisected there at.
Solution
Let OABC be a parallelogram Table O as origin. Let 
and 
be position vectors of 
such that 
Now,
Position vector of mid-point of D of A is
P.V. of a point dividing CD in the ratio
2 : 1 is
Again position vector of point divides OB in the ratio 1: 2 is
∴ position vectors of points trisecting CD and OB are same.
∴ CD trisects OB and CD is trisected there at.
and be position vectors of such that Now,
Position vector of mid-point of D of A is
P.V. of a point dividing CD in the ratio
2 : 1 is
Again position vector of point divides OB in the ratio 1: 2 is
∴ position vectors of points trisecting CD and OB are same.
∴ CD trisects OB and CD is trisected there at.
