Question
If P and Q are the mid-points of the sides AB and CD of a parallelogram ABCD, prove that DP and BQ cut the diagonal AC in its points of trisection which are also the points of trisection of DP and BQ respectively.
Solution
Take A as origin. Let 
be position vectors of B and D respectively such that
Now position vector of P, mid-point of AB, is
Also,
position vector of Q is 
Let E divide AC in the ratio 1 : 2 and F divide AC in the ratio 2 : 1
position vector of E is 
Position vector of point dividing PD in the ratio 1:2 is
Position vector of
is 
Position vector of point dividing BQ in the ratio 2 : 1 is
∴ DP and BQ cut diagonal AC in its points of trisection which are also the points of trisection of DP and BQ respectively.
be position vectors of B and D respectively such thatNow position vector of P, mid-point of AB, is
Also,
position vector of Q is Let E divide AC in the ratio 1 : 2 and F divide AC in the ratio 2 : 1
position vector of E is Position vector of point dividing PD in the ratio 1:2 is
Position vector of
is Position vector of point dividing BQ in the ratio 2 : 1 is
∴ DP and BQ cut diagonal AC in its points of trisection which are also the points of trisection of DP and BQ respectively.
