ABCD is any quadrilateral, P and Q are the midpoints of the diagonals AC & BD. M is the midpoint of PQ. If O is any point, prove that...?

P is the midpoint of AC , so
OP = (1/2)OA + (1/2)OC ... (1)

Q is the midpoint of BD , so
OQ = (1/2)OB + (1/2)OD ... (2)

M is the midpoint of PQ , so
OM
= (1/2)OP + (1/2)OQ
= (1/2)*[ (1/2)OA + (1/2)OC ] + (1/2)*[ (1/2)OB + (1/2)OD ] , by (1) & (2)
= (1/4)( OA + OC + OB + OD )

Thus
4OM = OA + OC + OB + OD

Q.E.D.