Let D, E, F are the midpoints of the sides `bar(BC), bar(CA) " and " bar(AB) ` of the triangle ABC. Prove that `vec(AD)+vec(BE)+vec(CF)=vec(0)`.

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) .

If G be the centroid of the triangle ABC, then prove that vec(GA)+vec(GB)+vec(GC)=vec(0).

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 traingle , show that bar(PA)+bar(PB)+bar(PC)= bar(PD)+bar(PE)+bar(PF) .

If D,E and F are the middle points of the sides BC,CA and AB of the DeltaABC , then AD+BE+CF is

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 .

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) .

bar(AC) " and " bar(BD) are the diagonals of the parallelogram ABCD. Prove that, vec(AC)+vec(BD)=2vec(BC) " and " vec(AC)-vec(BD)=2vec(AB)

Let vec(a),vec(b),vec (c ) be the positions vectors of the vertices of a triangle , prove that the area of the triangle is 1/2| vec(a) xx vec(b) + vec(b) xx vec(c)+ vec(c)xx vec(a)|

ABCD is a parallelogram and P is the midpoint of the side bar(BC) . Prove that bar(AC) " and " bar(DP) meet in a common point of trisection.

If the lengths of the sides vec(BC), vec(CA) and vec(AB) of the triangle ABC be a:b:c respectively and the lengths of perpendicular from the circumcenter upon vec(BC), vec(CA) and vec(AB) be x, y, z respectively, then prove that, a/x + b/y + c/z = (abc)/(4xyz) .