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In the given problem, we are told that \( G \) is the centroid of \( \triangle ABC \), and we are asked to find the length of \( GE \).

### Key Information:
- \( G \) is the centroid, which means it divides each median into a 2:1 ratio, with the longer segment being closer to the vertex.
- \( BG = 6 \), which means \( BG \) is two-thirds of the length of the median \( BE \).
- \( AF = 12 \), and \( AE = 15 \).

Let's solve this step by step:

1. **Centroid Properties:**
   - Since \( G \) is the centroid, it divides the median \( BE \) into two parts: \( BG \) (the part from \( B \) to \( G \)) and \( GE \) (the part from \( G \) to \( E \)).
   - The ratio of the centroid division is 2:1. Thus, \( BG = 2 \times GE \).

2. **Finding \( GE \):**
   Since \( BG = 6 \), we can use the 2:1 ratio to calculate \( GE \).
   \[
   BG = 2 \times GE \implies 6 = 2 \times GE
   \]
   Solving for \( GE \):
   \[
   GE = \frac{6}{2} = 3
   \]

Thus, the length of \( GE \) is **3 units**.

### Would you like further details or clarifications on any step? 

### Related Questions:
1. How do we find the coordinates of a centroid given the vertices of a triangle?
2. What is the significance of the 2:1 ratio in a centroid's division of medians?
3. How can we verify if a given point is the centroid of a triangle?
4. What are other properties of centroids in triangles?
5. How is the area of a triangle related to its centroid?

### Tip:
The centroid always divides each median into a 2:1 ratio, and it is also the center of mass of a triangle!

Point G is the centroid of triangle ABC. BG = 6, AF = 12, AE = 15. Find the length of GE.

Learn how to find the length of GE in triangle ABC where G is the centroid, and the centroid divides the medians in a 2:1 ratio. Given BG = 6 and AE = 15, this problem covers key geometric concepts involving centroids and triangle medians. Suitable for Grades 8-10.

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