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The question in the image asks for the **sum and the sum of squares of the direction cosines** of the line that passes through a point \((x, y, z)\) and the triangle's centroid with vertices at \((1, 0, 0)\), \((0, 1, 0)\), and \((0, 0, 1)\).

### Steps to Solve:

1. **Find the Centroid of the Triangle:**
   - The centroid \( G \) of a triangle with vertices \((x_1, y_1, z_1)\), \((x_2, y_2, z_2)\), and \((x_3, y_3, z_3)\) is given by:
     \[
     G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}, \frac{z_1 + z_2 + z_3}{3} \right)
     \]
   - Here, the vertices are \((1, 0, 0)\), \((0, 1, 0)\), and \((0, 0, 1)\).
   - Therefore, the centroid \( G \) is:
     \[
     G = \left( \frac{1+0+0}{3}, \frac{0+1+0}{3}, \frac{0+0+1}{3} \right) = \left( \frac{1}{3}, \frac{1}{3}, \frac{1}{3} \right)
     \]

2. **Direction Ratios of the Line:**
   - The direction ratios of the line passing through \( (x, y, z) \) and \( G \left( \frac{1}{3}, \frac{1}{3}, \frac{1}{3} \right) \) are:
     \[
     \left( x - \frac{1}{3}, y - \frac{1}{3}, z - \frac{1}{3} \right)
     \]

3. **Direction Cosines:**
   - Let the direction ratios be \( l = x - \frac{1}{3} \), \( m = y - \frac{1}{3} \), and \( n = z - \frac{1}{3} \).
   - The direction cosines \( l', m', n' \) are given by:
     \[
     l' = \frac{l}{\sqrt{l^2 + m^2 + n^2}}, \quad m' = \frac{m}{\sqrt{l^2 + m^2 + n^2}}, \quad n' = \frac{n}{\sqrt{l^2 + m^2 + n^2}}
     \]
   - The sum of direction cosines is:
     \[
     l' + m' + n' = \frac{l + m + n}{\sqrt{l^2 + m^2 + n^2}}
     \]
   - Given that \( l + m + n = x + y + z - 1 \), the sum of the direction cosines is:
     \[
     \frac{x + y + z - 1}{\sqrt{l^2 + m^2 + n^2}}
     \]

4. **Sum of Squares of Direction Cosines:**
   - The sum of squares of the direction cosines is:
     \[
     (l')^2 + (m')^2 + (n')^2 = \frac{l^2 + m^2 + n^2}{l^2 + m^2 + n^2} = 1
     \]

### Conclusion:
The correct answer is:
\[
\text{Option D: } \frac{x+y+z-1}{\sqrt{(x-\frac{1}{3})^2 + (y-\frac{1}{3})^2 + (z-\frac{1}{3})^2}} = 1
\]

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Do you want more details on any of these steps or have any other questions?

### Related Questions:
1. What are direction cosines and how are they used in 3D geometry?
2. How do you find the centroid of more complex shapes or polygons?
3. What is the significance of the sum of squares of direction cosines always being equal to 1?
4. How do direction ratios differ from direction cosines?
5. Can you explain how to find the equation of a line in 3D space?

### Tip:
Remember, the direction cosines always satisfy \( l^2 + m^2 + n^2 = 1 \), which is a key property in 3D geometry problems involving lines and planes.

Learn how to calculate the sum and sum of squares of direction cosines in 3D geometry. This problem involves finding direction cosines of a line passing through a point and the centroid of a triangle. Suitable for advanced high school students studying 3D geometry.

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