Because O is the center of gravity, OD = (1/3) AD = (1/2) ao.

(Let the midpoint of AC be E, which is connected with DE, so DE=( 1/2)AB, and DE is parallel to AB, and triangle ABO is similar to triangle o DE, so OD/OA=DE/AB= 1/2).

Vector OB= vector OD+ vector DB

Vector OC= vector OD+ vector DC

So: vector OB+ vector OC=2* vector OD+ vector DB+ vector DC.

= vector AO+ vector DB- vector DB

=-vector OA

So: vector OA+ vector OB+ vector OC= vector zero.