In this problem, point G is the centroid of triangle $\triangle ABC$, and we are tasked with finding the length of a segment within the triangle using the given information.

The centroid of a triangle divides each median into two segments: one-third of the median length from the vertex to the centroid, and two-thirds from the centroid to the midpoint of the opposite side.

We know the following from the diagram:

• $BG = 6$
• $AF = 12$
• $AE = 15$

To approach this:

1. First, let's find the lengths of the medians and apply the centroid property to solve for unknown segment lengths.

Would you like to go step-by-step with the solution for each segment or jump straight to the final answer for the segment you're asking about?