To find the centroid of triangle \( ABC \), we can use the formula for the centroid \( G \) of a triangle with vertices \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \):
\[ G\left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \]
Given the vertices of the triangle:
- \( A(-2, -3) \)
- \( B(1, 0) \)
- \( C(-2, 2) \)
We can substitute these coordinates into the centroid formula.
Calculating the x-coordinate of the centroid:
\[ x_G = \frac{-2 + 1 - 2}{3} = \frac{-3}{3} = -1 \]
Calculating the y-coordinate of the centroid:
\[ y_G = \frac{-3 + 0 + 2}{3} = \frac{-1}{3} \]
Thus, the coordinates of the centroid \( G \) are:
\[ G\left(-1, -\frac{1}{3}\right) \]
This calculation confirms that the medians of triangle \( ABC \) do indeed meet at a single point, which is the centroid \( G \).
In summary, the centroid of triangle \( ABC \) is:
\[ G\left(-1, -\frac{1}{3}\right) \]