OD = 1/2OB + 1/2OC

Let CF be the other median.

Prove that AD and BE's intersection, G divides AD in the ratio of 2:1.

Proof:

OD = 1/2OB + 1/2OC

OF = 1/2OA + 1/2OB

OD - OF = 1/2OC - 1/2OA

OD + 1/2OA = OF + 1/2OC

2OD + OA = 2OD + OC

2/3OD + 1/3OA = 2/3OF + 1/3OC

OG = OG