The centroid \( G \) of a triangle \( \triangle ABC \) is the point where the three medians intersect. The coordinates of the centroid can be calculated by taking the average of the coordinates of the vertices of the triangle.
Let the coordinates of the vertices \( A \), \( B \), and \( C \) be given by:
- \( A(x_1, y_1) \)
- \( B(x_2, y_2) \)
- \( C(x_3, y_3) \)
The coordinates of the centroid \( G \) are given by the following formulas:
\[ G_x = \frac{x_1 + x_2 + x_3}{3} \]
\[ G_y = \frac{y_1 + y_2 + y_3}{3} \]
Thus, the coordinates of the centroid \( G \) are:
\[ G\left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \]
This means the centroid divides each median in a ratio of \( 2:1 \), with the longer segment being closer to the vertex. Therefore, the centroid is a central point that balances the triangle based on its vertices' positions.
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