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P and Q are the points of trisection of the diagonal BD of a parallelogram ABCD. Prove that CQ is parallel to AP. Prove also that AC bisects PQ.

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We know that, diagonals of a parallelogram bisect each other

OA = OC and OB = OD 

Since P and Q are point of intersection of BD 

BP = PQ = QD 

Now, OB = OD and BP = QD 

OB - BP = OD - QD 

OP = OQ 

Thus in quadrilateral APCQ, we have 

OA = OC and OP = OQ 

diagonals of quadrilateral APCQ bisect each other 

APCQ is a parallelogram 

Hence AP || CQ

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