If A, B, C and D are four non-collinear points in the plane such that `bar(AD)+bar(BD)+bar(CD)=bar(0)`, then prove that the points D is the centroid of the triangle ABC.

Explanation

Let the position vector of A,B,C are `veca,vecbandvecc` respectively.

Here position vector of `D=vecd`
Then , Positon vector of `vec(AD)=vecd-veca`
Position vector of `vec(BD)=vecd-vecb`
Position vector of `vec(CD)=vecd-vecc`
`vec(AD)+vec(BD)+vec(CD)=0 " " ["Given"]`
`vecd-veca+vecd-vecb+vecd-vecc=0`
`3vecd=veca+vecb+vecc`
`vecd=(veca+vecb+vecc)/(3)`
Hence, `vecd` is the centroid of `DeltaABC`.