Question

prove that the centroid of a triangle always divides the median in the ratio 2:1

Answer 1

Reflect the triangle along AC, you can get a diagram below:
ABCB 1 is a parallelogram.
BEB 1 is a straight line .
Since CD = AD 1 and CD // AD1 ,
DCD1 A is a parallelogram. (opposite sides equal and parallel.)
\ DG // CG1
Since BD = DC and DG // CG1 \ BG = GG 1 (intercept theorem)
BG : GG 1 = 1 : 1
Since GE = EG 1 , BG : GE = 2 : 1.