Question

Prove by vector method , that the medians of a triangle are concurrent. 

Answer 1

Let us consider a triangle ABC. Let O be the fixed point, 

Position vector of the mid point P is vector OP = ½ vector (OB + OC) = ½ vector( b + c )

 If G divides vector AP in the ratio 2 : 1

Then, the position vector of G =

The symmetry of this result show that , the point which divides the other two medians in the ratio 2 : 1 will also have the same position vector 

Hence, the medians of a triangle are concurrent at G.