If `z_1,z_2,z_3` are three complex numbers such that `|z_1|=|z_2|=1`, find the maximum value of `|z_1-z_2|^2+|z_2-z_3|^2+|z_3+z_1|^2`

Options

A. If arg`(z_1/z_2) = pi/2` then arg `((z-z_1)/(z-z_2)) gt pi/4` where |z| gt 1
B. `|z_1z_2 + z_2z_3 + z_3z_1 | = | z_1 + z_2 + z_3| `
C. `lm(((z_1+z_2)(z_2+z_3)(z_3+z_1))/(z_1.z_2.z_3))=0`
D. If `|z_1-z_2|=sqrt2|z_1-z_3| =sqrt2 |z_2-z_3| `, then Re`((z_3-z_1)/(z_3-z_2))=0`

Answer

To solve the problem, we need to find the maximum value of the expression \( |z_1 - z_2|^2 + |z_2 - z_3|^2 + |z_3 + z_1|^2 \) given that \( |z_1| = |z_2| = 1 \).

### Step-by-Step Solution:

1. **Understanding the Magnitude Condition**:
Since \( |z_1| = |z_2| = 1 \), we can represent \( z_1 \) and \( z_2 \) as points on the unit circle in the complex plane:
\[ z_1 = e^{i\theta_1}, \quad z_2 = e^{i\theta_2} \]
where \( \theta_1 \) and \( \theta_2 \) are angles.

2. **Expressing the Distances**:
The expression we need to maximize can be rewritten using the properties of complex numbers:
\[ |z_1 - z_2|^2 = (z_1 - z_2)(\overline{z_1 - z_2}) = |z_1|^2 + |z_2|^2 - 2 \text{Re}(z_1 \overline{z_2}) = 1 + 1 - 2 \text{Re}(z_1 \overline{z_2}) = 2 - 2 \cos(\theta_1 - \theta_2) \]

3. **Calculating Each Term**:
Similarly, we can express \( |z_2 - z_3|^2 \) and \( |z_3 + z_1|^2 \):
\[ |z_2 - z_3|^2 = 2 - 2 \cos(\theta_2 - \theta_3) \]
\[ |z_3 + z_1|^2 = |z_3|^2 + |z_1|^2 + 2 \text{Re}(z_3 \overline{z_1}) = 1 + 1 + 2 \text{Re}(z_3 \overline{z_1}) = 2 + 2 \cos(\theta_3 + \theta_1) \]

4. **Combining the Terms**:
Now, we combine all these expressions:
\[ |z_1 - z_2|^2 + |z_2 - z_3|^2 + |z_3 + z_1|^2 = (2 - 2 \cos(\theta_1 - \theta_2)) + (2 - 2 \cos(\theta_2 - \theta_3)) + (2 + 2 \cos(\theta_3 + \theta_1)) \]
This simplifies to:
\[ 6 - 2(\cos(\theta_1 - \theta_2) + \cos(\theta_2 - \theta_3)) + 2 \cos(\theta_3 + \theta_1) \]

5. **Maximizing the Expression**:
To maximize the expression, we need to minimize \( \cos(\theta_1 - \theta_2) + \cos(\theta_2 - \theta_3) \) and maximize \( \cos(\theta_3 + \theta_1) \). The maximum value of \( \cos \) is 1 and the minimum value is -1.

If we set \( \theta_1 = 0 \), \( \theta_2 = \pi \) (180 degrees), and \( \theta_3 = 0 \), we can achieve:
\[ \cos(\theta_1 - \theta_2) = -1, \quad \cos(\theta_2 - \theta_3) = -1, \quad \cos(\theta_3 + \theta_1) = 1 \]

6. **Calculating the Maximum Value**:
Plugging these values into our expression:
\[ 6 - 2(-1 - 1) + 2(1) = 6 + 4 + 2 = 12 \]

7. **Conclusion**:
Therefore, the maximum value of \( |z_1 - z_2|^2 + |z_2 - z_3|^2 + |z_3 + z_1|^2 \) is:
\[ \boxed{12} \]