Three vertices of a parallelogram are (0, 0, 0), (2, -1, 2) and (5, 6, 8) Find the coordinates of
the Fourth vertex.

Answer

To find the coordinates of the fourth vertex of the parallelogram given three vertices A(0, 0, 0), B(2, -1, 2), and C(5, 6, 8), we can use the property that the diagonals of a parallelogram bisect each other.

### Step-by-Step Solution:

1. **Identify the Given Points:**
- Let A = (0, 0, 0)
- Let B = (2, -1, 2)
- Let C = (5, 6, 8)
- Let D = (x, y, z) be the fourth vertex we need to find.

2. **Find the Midpoint of Diagonal AC:**
The midpoint \( R \) of diagonal \( AC \) can be calculated using the midpoint formula:
\[ R = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right) \]
For points A and C:
- \( x_1 = 0, y_1 = 0, z_1 = 0 \)
- \( x_2 = 5, y_2 = 6, z_2 = 8 \)

Thus, the coordinates of midpoint \( R \) are:
\[ R = \left( \frac{0 + 5}{2}, \frac{0 + 6}{2}, \frac{0 + 8}{2} \right) = \left( \frac{5}{2}, 3, 4 \right) \]

3. **Set Up the Midpoint of Diagonal BD:**
Since the diagonals bisect each other, the midpoint of diagonal \( BD \) must also be \( R \):
\[ R = \left( \frac{x + 2}{2}, \frac{y - 1}{2}, \frac{z + 2}{2} \right) \]

4. **Equate the Midpoints:**
Now we can set up equations based on the coordinates of midpoint \( R \):
- For the x-coordinates:
\[ \frac{x + 2}{2} = \frac{5}{2} \]
- For the y-coordinates:
\[ \frac{y - 1}{2} = 3 \]
- For the z-coordinates:
\[ \frac{z + 2}{2} = 4 \]

5. **Solve for x, y, and z:**
- From the x-coordinate equation:
\[ x + 2 = 5 \implies x = 5 - 2 = 3 \]
- From the y-coordinate equation:
\[ y - 1 = 6 \implies y = 6 + 1 = 7 \]
- From the z-coordinate equation:
\[ z + 2 = 8 \implies z = 8 - 2 = 6 \]

6. **Conclusion:**
The coordinates of the fourth vertex \( D \) are:
\[ D = (3, 7, 6) \]

### Final Answer:
The coordinates of the fourth vertex of the parallelogram are \( D(3, 7, 6) \).