Let `A B C D` be a p[arallelogram whose diagonals intersect at `P` and let `O` be the origin. Then prove that ` vec O A+ vec O B+ vec O C+ vec O D=4 vec O Pdot`

Explanation

Since the diagonals of a parallelogram bisect each other, P is the middle point of AC and BD both. Therefore,
`vec(OA) + vec(OC) = 2 vec(OP) and vec(OB) + vec(OD) = 2 vec(OP)`