Three vertices of a parallelogram ABCD are A (3,-1,2), B (1, 2, 4) and C(-1,1,2). Find the coordinates of the fourth vertex D.

Answer

To find the coordinates of the fourth vertex \( D \) of the parallelogram \( ABCD \) given the coordinates of vertices \( A \), \( B \), and \( C \), we can use the property that the diagonals of a parallelogram bisect each other.

### Step-by-Step Solution:

1. **Identify the Coordinates of Points**:
- Let the coordinates of point \( A \) be \( A(3, -1, 2) \).
- Let the coordinates of point \( B \) be \( B(1, 2, 4) \).
- Let the coordinates of point \( C \) be \( C(-1, 1, 2) \).
- Let the coordinates of point \( D \) be \( D(x_4, y_4, z_4) \).

2. **Use the Midpoint Formula**:
- The midpoint of diagonal \( AC \) can be calculated using the formula:
\[ \text{Midpoint of } AC = \left( \frac{x_1 + x_3}{2}, \frac{y_1 + y_3}{2}, \frac{z_1 + z_3}{2} \right) \]
- Substituting the coordinates of \( A \) and \( C \):
\[ \text{Midpoint of } AC = \left( \frac{3 + (-1)}{2}, \frac{-1 + 1}{2}, \frac{2 + 2}{2} \right) = \left( \frac{2}{2}, \frac{0}{2}, \frac{4}{2} \right) = (1, 0, 2) \]

3. **Calculate the Midpoint of Diagonal \( BD \)**:
- The midpoint of diagonal \( BD \) can be calculated using the formula:
\[ \text{Midpoint of } BD = \left( \frac{x_2 + x_4}{2}, \frac{y_2 + y_4}{2}, \frac{z_2 + z_4}{2} \right) \]
- Substituting the coordinates of \( B \) and \( D \):
\[ \text{Midpoint of } BD = \left( \frac{1 + x_4}{2}, \frac{2 + y_4}{2}, \frac{4 + z_4}{2} \right) \]

4. **Set the Midpoints Equal**:
- Since the midpoints of diagonals \( AC \) and \( BD \) are equal, we can set them equal to each other:
\[ (1, 0, 2) = \left( \frac{1 + x_4}{2}, \frac{2 + y_4}{2}, \frac{4 + z_4}{2} \right) \]

5. **Create Equations from the Equal Midpoints**:
- From the x-coordinates:
\[ 1 = \frac{1 + x_4}{2} \implies 2 = 1 + x_4 \implies x_4 = 1 \]
- From the y-coordinates:
\[ 0 = \frac{2 + y_4}{2} \implies 0 = 2 + y_4 \implies y_4 = -2 \]
- From the z-coordinates:
\[ 2 = \frac{4 + z_4}{2} \implies 4 = 4 + z_4 \implies z_4 = 0 \]

6. **Conclusion**:
- Therefore, the coordinates of point \( D \) are:
\[ D(1, -2, 0) \]

### Final Answer:
The coordinates of the fourth vertex \( D \) are \( (1, -2, 0) \).