ABCD is a parallelogram and P is a point of intersection of lines joining the mid points of the sides. 0 is any point in the plane of the parallelogram. Show that : OA+ OB + OC + OD = 4OP .

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Solution

P is a mid point of AC and BD
$- - \to O P = \frac{- - \to O A + - - \to O C}{2}$
$20 \to P = - - \to O A + - - \to O C$
and $- - \to O P = \frac{- - \to O B + - - \to O D}{2}$
$- - \to O B + - - \to O D = 2 - - \to O P$
add above equations $- - \to O A + - - \to O B + - - \to O C + - - \to O D = 4 - - \to O P$

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