Where is B located? I will need to assume that B is located at the midpoint of side AC, in order to solve the problem.
Under this assumption, BE is a median. Along any median, the centroid of a triangle is always located 2/3 of the way from the vertex to the midpoint of the opposite side, so the centroid G is 2/3 of the way from E to B.
So we have GE = (2/3)BE.
Furthermore, by segment addition, BG + GE = BE.
Since BE = 9, GE = (2/3)(9) = 6.
Then BG + 6 = 9, which gives BG = 3.
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