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