If `A=(1, 2, 3), B=(4, 5, 6), C=(7, 8, 9)` and D, E, F are the mid points of the triangle ABC, then find the centroid of the triangle DEF.

Answer

To find the centroid of triangle DEF, where D, E, and F are the midpoints of triangle ABC with vertices A(1, 2, 3), B(4, 5, 6), and C(7, 8, 9), we will follow these steps:

### Step 1: Find the midpoints D, E, and F of triangle ABC.

1. **Calculate D (midpoint of AB)**:
\[ D = \left( \frac{x_A + x_B}{2}, \frac{y_A + y_B}{2}, \frac{z_A + z_B}{2} \right) = \left( \frac{1 + 4}{2}, \frac{2 + 5}{2}, \frac{3 + 6}{2} \right) = \left( \frac{5}{2}, \frac{7}{2}, \frac{9}{2} \right) \]

2. **Calculate E (midpoint of AC)**:
\[ E = \left( \frac{x_A + x_C}{2}, \frac{y_A + y_C}{2}, \frac{z_A + z_C}{2} \right) = \left( \frac{1 + 7}{2}, \frac{2 + 8}{2}, \frac{3 + 9}{2} \right) = \left( \frac{8}{2}, \frac{10}{2}, \frac{12}{2} \right) = (4, 5, 6) \]

3. **Calculate F (midpoint of BC)**:
\[ F = \left( \frac{x_B + x_C}{2}, \frac{y_B + y_C}{2}, \frac{z_B + z_C}{2} \right) = \left( \frac{4 + 7}{2}, \frac{5 + 8}{2}, \frac{6 + 9}{2} \right) = \left( \frac{11}{2}, \frac{13}{2}, \frac{15}{2} \right) \]

### Step 2: Find the centroid of triangle DEF.

The formula for the centroid \(G\) of a triangle with vertices \(D(x_1, y_1, z_1)\), \(E(x_2, y_2, z_2)\), and \(F(x_3, y_3, z_3)\) is given by:
\[ G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}, \frac{z_1 + z_2 + z_3}{3} \right) \]

Substituting the coordinates of points D, E, and F:
- \(D = \left( \frac{5}{2}, \frac{7}{2}, \frac{9}{2} \right)\)
- \(E = (4, 5, 6)\)
- \(F = \left( \frac{11}{2}, \frac{13}{2}, \frac{15}{2} \right)\)

#### Calculate the x-coordinate of G:
\[ x_G = \frac{\frac{5}{2} + 4 + \frac{11}{2}}{3} = \frac{\frac{5 + 8 + 11}{2}}{3} = \frac{\frac{24}{2}}{3} = \frac{12}{3} = 4 \]

#### Calculate the y-coordinate of G:
\[ y_G = \frac{\frac{7}{2} + 5 + \frac{13}{2}}{3} = \frac{\frac{7 + 10 + 13}{2}}{3} = \frac{\frac{30}{2}}{3} = \frac{15}{3} = 5 \]

#### Calculate the z-coordinate of G:
\[ z_G = \frac{\frac{9}{2} + 6 + \frac{15}{2}}{3} = \frac{\frac{9 + 12 + 15}{2}}{3} = \frac{\frac{36}{2}}{3} = \frac{18}{3} = 6 \]

### Final Result:
The coordinates of the centroid \(G\) of triangle DEF are:
\[ G = (4, 5, 6) \]