G is centroid of `Delta ABC` and a,b,c are the lengths of the sides BC, CA and AB respectively. Prove that `a^(2)+b^(2)+c^(2) =3(bar(OA)^(2)+bar(OB)^(2)+bar(OC)^(2))-9(bar(OG))^(2)` where O is any point.

If D is the midpoint of the side BC of a triangle ABC then bar(AB)^(2)+bar(AC)^(2)=

If A,B,C,D are four points in space satisfying bar(AB).bar(CD)=k[|bar(AD)|^(2)+|bar(BC)|^(2)-|bar(AC)|^(2)-|bar(BD)|^(2)] then k =

In Delta ABC, if bar(a), bar(b), bar(c) are position vectors of the vertices A, B, and C respectively, then prove that the position vector of the centroid G is (1)/(3) (bar(a) + bar(b) + bar(c))

A point I is the centre of a circle inscribed in a triangle ABC then show that abs(bar(BC))bar(IA)+abs(bar(CA))bar(IB)+abs(bar(AB))bar(IC)=bar(0)

Find the vector equation of the line through the centroid of triangle ABC and parallel to side BC where position vector of A, B, C are respectively are bar(i)-2bar(j)+bar(k), 2bar(i)-bar(j), bar(i)+bar(j)+3bar(k) .

The points D, E, F are the midpoints of the sides bar(BC), bar(CA),bar(AB) " of " Delta ABC respectively. If A=(-2,3,4), D =(1,-4,2), E =(-5,2,-3) then F=

Show that (bar(a)+bar(b)) . [(bar(b)+bar(c)) xx (bar(c )+bar(a))] = 2[bar(a)bar(b)bar(c )] .