If `z_(1),z_(2),z_(3)` are the vertices of an isoscles triangle right angled at `z_(2)`, then

Options

A. `z_(1)^(2)+2z_(2)^(2)+z_(3)^(2)=2z_(2)(z_(1)+z_(3))`
B. `z_(1)^(2)+z_(2)^(2)+z_(3)^(2)=2z_(2)(z_(1)+z_(3))`
C. `z_(1)^(2)+z_(2)^(2)+2z_(3)^(2)=2z_(2)(z_(1)+z_(3))`
D. `2z_(1)^(2)+z_(2)^(2)+z_(3)^(2)=2z_(2)(z_(1)+z_(3))`

Explanation

To solve the problem, we need to establish the relationship between the vertices of the isosceles triangle right-angled at \( z_2 \). Let's denote the vertices of the triangle as \( z_1, z_2, z_3 \).

### Step-by-Step Solution:

1. **Understanding the Triangle Configuration**:
- We have an isosceles triangle \( \triangle z_1 z_2 z_3 \) with the right angle at \( z_2 \). This means that the lengths of the sides \( z_1 z_2 \) and \( z_2 z_3 \) are equal.
- Therefore, we can write:
\[ |z_1 - z_2| = |z_3 - z_2| \]

2. **Using the Right Angle Condition**:
- Since the triangle is right-angled at \( z_2 \), we can use the property of right triangles in the complex plane. The vectors \( z_1 - z_2 \) and \( z_3 - z_2 \) are perpendicular.
- This means that the dot product of these vectors must equal zero:
\[ \text{Re}((z_1 - z_2) \overline{(z_3 - z_2)}) = 0 \]

3. **Expressing the Vectors**:
- Let \( z_1 - z_2 = a \) and \( z_3 - z_2 = b \). The condition of equal lengths gives us:
\[ |a| = |b| \]
- The perpendicularity condition gives:
\[ \text{Re}(a \overline{b}) = 0 \]

4. **Using Polar Form**:
- We can express \( a \) and \( b \) in polar form. Since they are equal in magnitude and perpendicular, we can write:
\[ b = a e^{i \frac{\pi}{2}} \quad \text{or} \quad b = a e^{-i \frac{\pi}{2}} \]
- This implies:
\[ z_3 - z_2 = (z_1 - z_2) i \quad \text{or} \quad z_3 - z_2 = (z_1 - z_2)(-i) \]

5. **Setting Up the Equation**:
- From the above expressions, we can derive:
\[ z_3 = z_2 + i(z_1 - z_2) \quad \text{or} \quad z_3 = z_2 - i(z_1 - z_2) \]

6. **Squaring Both Sides**:
- If we take the first case, we can express the relationship:
\[ |z_3 - z_2|^2 = |z_1 - z_2|^2 \]
- This leads to:
\[ (z_3 - z_2)(\overline{z_3 - z_2}) = (z_1 - z_2)(\overline{z_1 - z_2}) \]

7. **Final Equation**:
- After manipulating the equations, we can arrive at the final relationship:
\[ |z_1 - z_2|^2 + |z_3 - z_2|^2 = 2|z_2 - \frac{(z_1 + z_3)}{2}|^2 \]

### Conclusion:
The relationship between the vertices of the isosceles triangle right-angled at \( z_2 \) can be expressed as:
\[ |z_1 - z_2|^2 + |z_3 - z_2|^2 = 2|z_2 - \frac{(z_1 + z_3)}{2}|^2 \]