`A B C D` is a quadrilateral and `E` and the point intersection of the lines joining the middle points of opposite side. Show that the resultant of ` vec O A , vec O B , vec O Ca n d vec O D` is equal to 4 ` vec O E ,` where `O` is any point.

Explanation

We know that the figure formed by the lines joining the midpoints of the sides of a qudrilateral is a parallelogram. Hence, MPNQ is a parallelogramm, whose diagonals are MN and PQ intersecting at E, which is the midpoint of both MN and PQ. For any origin O, we have `vec(OA) + vec(OB) = 2(vec(OM))` (as M is the midpoint of AB)

`vec(OC) +vec(OB) = 2(vec(ON))` (as N is the midpoint of BC)
`rArr vec(OA) + vec(OB) + vec(OC) + vec(OD) = 2 (vec(OM) + vec(ON))`
`" " = 2(2 vec(OE)) = 4vec(OE)`
Where E is the midpoint of MN as it is the intersection of the diagonals of a parallelogram.