The vertices of a square are `z_1,z_2,z_3 and z_4` taken in the anticlockwise order, then `z_3=`

Options

A. `-iz_(1)+(1+i)z_(2)`
B. `iz_(1)+(1-i)z_(2)`
C. `z_(1)+(1+i)z_(2)`
D. `(1+i)z_(1)+z_(2)`

Explanation

To solve the problem of finding \( z_3 \) given the vertices of a square \( z_1, z_2, z_3, z_4 \) in anticlockwise order, we can follow these steps:

### Step-by-step Solution:

1. **Understanding the Properties of a Square**:
The diagonals of a square bisect each other. Therefore, we can use the property that the midpoint of the diagonals is the same.

2. **Setting Up the Equation**:
Since the diagonals bisect each other, we have:
\[ \frac{z_1 + z_3}{2} = \frac{z_2 + z_4}{2} \]
Multiplying both sides by 2 gives us:
\[ z_1 + z_3 = z_2 + z_4 \quad \text{(Equation 1)} \]

3. **Rotation of Points**:
The vertices of the square can be obtained by rotating the points. The line \( AD \) can be obtained by rotating the line \( AB \) through an angle of \( \frac{\pi}{2} \) (90 degrees). This gives us the relationship:
\[ z_4 - z_1 = (z_2 - z_1) e^{i\frac{\pi}{2}} \]
Since \( e^{i\frac{\pi}{2}} = i \), we can rewrite this as:
\[ z_4 - z_1 = i(z_2 - z_1) \]

4. **Rearranging the Equation**:
Rearranging the above equation gives:
\[ z_4 = z_1 + i(z_2 - z_1) = z_1 + iz_2 - iz_1 = (1 - i)z_1 + iz_2 \]

5. **Substituting into Equation 1**:
Now substitute \( z_4 \) back into Equation 1:
\[ z_1 + z_3 = z_2 + ((1 - i)z_1 + iz_2) \]
Simplifying this gives:
\[ z_1 + z_3 = z_2 + (1 - i)z_1 + iz_2 \]
\[ z_1 + z_3 = (1 + i)z_2 + (1 - i)z_1 \]

6. **Solving for \( z_3 \)**:
Rearranging to isolate \( z_3 \):
\[ z_3 = (1 + i)z_2 + (1 - i)z_1 - z_1 \]
\[ z_3 = (1 + i)z_2 + (1 - 2i)z_1 \]

### Final Result:
Thus, we have derived that:
\[ z_3 = (1 + i)z_2 + (1 - 2i)z_1 \]