Prove that centroid of a triangle divides its median in the ratio of 2:1

Let ABC be the given triangle and AD, BE and CF be the respective medians and let G be its centroid.

Construction: Produce AD to O such that AG = GO

Now, In ∆ AGE and ∆ AOC

∠ GAE = ∠ OAC (Common)

So ∆ AGE ~ ∆ AOC (by SAS similarly criterion)

Thus, ∠ AGE = ∠ AOC and GO is the transversal 

⇒ GE ║ OC

⇒ BE ║ OC

⇒ BG ║ OC  ........ (1)

Similarly

GC ║ BO  ....... (2)

from (1) and (2) we can conclude that

BGCO is a parallelogram

also we know that diagonals of a parallelogram bisect each other.

⇒ GD = DO

Now AG = GO = GD + DO = GD + GD = 2 GD

⇒AG : GD = 2 : 1

Similarly BG : GE = CG : GF = 2 : 1

Hence the centroid of a triangle divides it in the ratio 2 : 1