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

Explanation

Let ABC be a triangle and D,E,F be the mid points of the sides of `triangleABC`
Let `veca,vecb` and `vecc` be the position vectors of the points A, B, C. Then the position vectors of D< E, F are
`(vecb+vecc)/2, (vecc+veca)/2` and `(veca+vecb)/2` respectively.
Let G be the point which divides the median into the ratio 2:1.
Then the position vector of G is
`2((veca+vecb)/2+1veca)/(2+1)` i.e. `(veca+vecb+vecc)/3`
The summetry of the result shows that the point G also lies on the other two medians.
Hence the medians are concurrent. (Proved)