Grade 12 question - vector proof (Centroids)

[eekoz]

Okay, so I have a triagle with vertices A, B, and C.
I know that the centroid, G, is where all the medians of the triangle intersect, and G divides the median at a 2:1 ratio

Assuming point O is a point that's not on the triangle, how can I prove:
OG = 1/3(OA + OB + OC) ?

Thanks in advance!

 

[Gene]

Suppose it is a 45° triangle with legs of 10 and the right angle at C = (0,0).
The centroid is at (10/3,10/3)
Further suppose that O is VERY close to C.
OA = 10
OB = 10
OC = ~0
OG=sqrt(2*(10/3)²)=10sqrt(2)/3
(OA+OB+OC)/3 = 20/3
Now all you have to prove is that
sqrt(2)=2
It might be easier to prove that
OG=1/3sqrt(OA²+OB²+OC²)