To solve the problem, we need to determine the value of $x$ given the conditions:

• $BE = 9x - 10$
• $EC = 5x + 2$

Key Information:

1. G is the circumcenter:
   • In a triangle, the circumcenter is equidistant from all the vertices.
   • This implies that $BE = EC$, as they are radii of the circumcircle.

Step-by-step Solution:

1. Set the given expressions for $BE$ and $EC$ equal: $9x - 10 = 5x + 2$

2. Simplify the equation: Subtract $5x$ from both sides: $4x - 10 = 2$

   Add $10$ to both sides: $4x = 12$

3. Solve for $x$: Divide both sides by 4: $x = 3$

Final Answer:

The value of $x$ is $\boxed{3}$.

Would you like me to verify the solution, or do you have additional questions?

Here are 5 related questions to expand your understanding:

1. What is the significance of the circumcenter in a triangle?
2. How do you find the circumcenter in different types of triangles?
3. Can the circumcenter lie outside the triangle? If so, in what cases?
4. What are some real-world applications of circumcenters in geometry?
5. How would the problem change if the triangle were scalene or obtuse?

Tip: Always ensure your geometric conditions (like equidistant properties) are applied correctly when solving such problems!