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ABCD is a parallelogram and P is the point of intersection of its diagonals. If O is the origin of reference, show that

\(\vec {OA}+\vec {OB}+\vec{OC}+\vec{OD}=4\vec{OP}.\)

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Let position vectors of the vertices A, B, C and D of the parallelogram ABCD with respect to O be \(\vec a,\vec b,\vec c\)and \(\vec d\) and respectively.

Also, let us assume position vector of P is \(\vec p\).

Given ABCD is a parallelogram. We know that the two diagonals of a parallelogram bisect each other. So, P is the midpoint of AC and BD.

As P is the midpoint of AC, using midpoint formula,

we have

P is also the midpoint of BC.

So,

Now we have

Adding these two equations, we get

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