Question

if p and q are points on any trisection of the diagonal BD of a parallelogram ABCD prove that cQ is parallel to AP ​

Answer 1

Given ABCD is a parallelogram. AC and BD are the diagonals of the parallelogram which intersect at point O.

The points P and Q trisects the diagonal BD.

Now, since the diagonals of a parallelogram bisect each other

So, Ac and Bd bisect each other at O

=> OB = OD and OA = OC

Now, P and PQ trisect the diagonal BD

So, DQ = PQ = BP

OB = OD and BP = DQ

Now, OB - BP = OD - DQ

=> OP = OQ

Thus, in quadrilateral APCQ, diagonal AC and PQ are such that OP = OQ and OA = OC

So, the diagonal AC and PQ bisect each other.

Again if the diagonal of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

Hence, ABCD is a parallelogram.

So, CQ || AP {since sides of parallelogram are parallel}