Question

Let G be the centroid of triangle ABC. Simply the vector sum GA+GB+GC.

Answer 1

Perhaps not a 'proof,' but more of an experiment to get an idea of what happens in the situation:
http://puu.sh/1yzIG
I simply played with a picture from Wikipedia's article on Centroids (zoomed in for clarity) and labeled some things / moved some lines around.

It seems we will get the zero vector as all three vectors cancel.

Answer 2

I suppose one way to actually prove it is to use coordinate geometry.

We can find the centroid as the intersection point of the medians. So, if we used some arbitrary triangle ABC with coordinates A(0,0), B(b,0), and C(c,d); then we would be able to find the midpoints of each side and G as the intersection of each vertex and the opposite midpoint, and then the vectors GA, GB, and GC all with some Geometry work.