Let ABCD be a quadrilateral. Let E be the point of intersection of the line joining the midpoints of the opposite sides. If O is any point and  OA→+OB→+OC→+OD→=k OE→ then is k equal to

Let OA→=a→,OB→=b→,OC→=c→ and OD→=d→Therefore, OA→+OB→+OC→+OD→=a→+b→+c→+d→P, the midpoint of AB ,is a→+b→2Q, the midpoint of  CD→, is c→+d→2Therefore, the midpoint of  PQ→ is a→+b→+c→+d→4Similarly, the midpoint of RS is a→+b→+c→+d→4 i.e., OE→=a→+b→+c→+d→4⇒k=4.