This image provides a solution for finding the coordinates of the centroid of a triangle in 3D space. The vertices of the triangle are given as points $A(x_1, y_1, z_1)$, $B(x_2, y_2, z_2)$, and $C(x_3, y_3, z_3)$.

The centroid $G$ of a triangle is the point of intersection of the medians, and it divides each median into two segments in the ratio 2:1.

Steps:

1. The centroid (G) is calculated as the average of the coordinates of the three vertices: $G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}, \frac{z_1 + z_2 + z_3}{3} \right)$ This formula gives the coordinates of the centroid $G$ directly without needing to calculate the midpoint of the median.

2. Midpoint of BC (D): The point $D$ is the midpoint of $BC$, given by: $D = \left( \frac{x_2 + x_3}{2}, \frac{y_2 + y_3}{2}, \frac{z_2 + z_3}{2} \right)$ This point $D$ lies on the median $AD$, which connects vertex $A$ to the midpoint $D$ of side $BC$.

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