Question 1119150
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There is something wrong with the statement of the problem.  As you found, the radius of a circle inscribed in an equilateral triangle of side length 4 is 2sqrt3 divided by 3; but the problem says the radius is 1.<br>
Your definition of point G being "on the edge of the circle" is not precise.<br>
If G were a point on the circle where it is tangent to a side of the triangle, then clearly the distance between C and G would be half the side length of the triangle, which is 2.  So that is probably not where point G is.<br>
The only other reasonable place for point G is a point on the circle closest to a vertex of the triangle.  In that case, since the altitude of the triangle is 2sqrt3 and the radius of the circle is 2sqrt3 divided by 3, the distance from C to G is also 2sqrt3 divided by 3.<br>
So if you ignore the misinformation that the radius of the circle is 1, your answer of 2sqrt3 divided by 3 for the distance from C to G is correct.