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