Vector proof question

1. Apr 20, 2006

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) ?

I've seen this equation a lot of times, but I'd like to see a proof of it for interest sake

Thanks

2. Apr 21, 2006

AKG

Let $P$ be the midpoint of $\overline{BC}[/tex]. Then: $$\overline{OB} + \frac{1}{2}\overline{BC} = \overline{OP}$$ $$\overline{OB} + \overline{BC} = \overline{OC}$$ $$\overline{OP} + \overline{PA} = \overline{OA}$$ $$\overline{OP} + \frac{1}{3}\overline{PA} = \overline{OG}$$ The last line comes from the fact that G divides the median [itex]\overline{PA}$ at a 2:1 ratio. You should be able to figure it out from here.