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

Answer

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

Let's denote the coordinates of the fourth vertex \( D \) as \( (x, y, z) \).

### Step 1: Find the midpoint of diagonal \( AC \)

The coordinates of points \( A \) and \( C \) are:
- \( A(1, 2, 1) \)
- \( C(1, 6, 0) \)

Using the midpoint formula, the midpoint \( O \) of \( AC \) is given by:
\[ O = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right) \]
Substituting the coordinates of \( A \) and \( C \):
\[ O = \left( \frac{1 + 1}{2}, \frac{2 + 6}{2}, \frac{1 + 0}{2} \right) = \left( 1, 4, \frac{1}{2} \right) \]

### Step 2: Use the midpoint of diagonal \( BD \)

Since \( O \) is also the midpoint of diagonal \( BD \), we can set up the equations using the coordinates of point \( B(2, 5, 6) \) and \( D(x, y, z) \):
\[ O = \left( \frac{2 + x}{2}, \frac{5 + y}{2}, \frac{6 + z}{2} \right) \]
Setting this equal to the coordinates of \( O \):
\[ \left( \frac{2 + x}{2}, \frac{5 + y}{2}, \frac{6 + z}{2} \right) = \left( 1, 4, \frac{1}{2} \right) \]

### Step 3: Solve for \( x, y, z \)

From the equations, we can derive three separate equations:

1. From the x-coordinates:
\[ \frac{2 + x}{2} = 1 \implies 2 + x = 2 \implies x = 0 \]

2. From the y-coordinates:
\[ \frac{5 + y}{2} = 4 \implies 5 + y = 8 \implies y = 3 \]

3. From the z-coordinates:
\[ \frac{6 + z}{2} = \frac{1}{2} \implies 6 + z = 1 \implies z = -5 \]

### Step 4: Conclusion

Thus, the coordinates of the fourth vertex \( D \) are:
\[ D(0, 3, -5) \]