ABCD is a quadrilateral and E is the point of intersection of the lines joining the mid-points of opposite sides. If O be any point in the plane, then show that `vec(OA)+vec(OB)+vec(OC)+vec(OD)=4vec(OE)`.

ABCD is a quadrilateral and E is the point of intersection of the lines joining the middle points of opposite side. Show that the resultant of vec (OA) , vec(OB) , vec(OC) and vec(OD) = 4 vec(OE) , where O is any point.

A B C D is a quadrilateral. E is the point of intersection of the line joining the midpoints of the opposite sides. If O is any point and vec O A+ vec O B+ vec O C+ vec O D=x vec O E ,t h e nx is equal to a. 3 b. 9 c. 7 d. 4

C is the midpoint of the line segment bar(AB) and O is any point outside AB, show that vec(OA)+vec(OB)=2vec(OC) .

Show that the straight lines joining the mid - points of the opposite sides of a quadrilateral bisect each other .

D, E, F are the midpoints of the sides bar(BC), bar(CA) " and " bar(AB) respectively of the triangle ABC. If P is any point in the plane of the triangle, show that vec(PA)+vec(PB)+vec(PC)=vec(PD)+vec(PE)+vec(PF) .

The diagonals of the parallelogram ABCD intersect at E. If vec(a), vec(b), vec(c) " and " vec(d) be the position vectors of its vertices with respect to an arbitary origin O, then show that vec(a)+vec(b)+vec(c)+vec(d)=4vec(OE) .

Let D ,Ea n dF be the middle points of the sides B C ,C Aa n dA B , respectively of a triangle A B Cdot Then prove that vec A D+ vec B E+ vec C F= vec0 .

Let A B C D be a parallelogram 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

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

If A (veca).B(vecb) and C(vecc) are three non-collinear point and origin does not lie in the plane of the points A, B and C, then for any point P(vecP) in the plane of the triangleABC such that vector vec(OP) is bot to plane of trianglABC , show that vec(OP)=([vecavecbvecc] (vecaxxvecb+vecbxxvecc+veccxxveca))/(4Delta^(2))