Let `A(z_(1)),B(z_(2)) and C(z_(3))` be complex numbers satisfying the equation`|z|=1` and also satisfying the relation `3z_(1)=2z_(2)+2z_(3)`. Then `|z_(2)-z_(3)|^(2)` is equal to

Answer

To solve the problem, we need to find the value of \(|z_2 - z_3|^2\) given the conditions that \(|z| = 1\) for complex numbers \(z_1, z_2, z_3\) and the relation \(3z_1 = 2z_2 + 2z_3\).

### Step-by-Step Solution:

1. **Understanding the Condition**:
The condition \(|z| = 1\) implies that \(z_1, z_2, z_3\) lie on the unit circle in the complex plane. This means that each of these complex numbers can be represented in the form \(z_k = e^{i\theta_k}\) where \(\theta_k\) is the angle corresponding to \(z_k\).

2. **Rearranging the Given Relation**:
We start with the equation:
\[ 3z_1 = 2z_2 + 2z_3 \]
Dividing both sides by 2 gives:
\[ \frac{3}{2} z_1 = z_2 + z_3 \]

3. **Finding the Midpoint**:
From the equation \(\frac{3}{2} z_1 = z_2 + z_3\), we can express \(z_2 + z_3\) as:
\[ z_2 + z_3 = \frac{3}{2} z_1 \]
This indicates that the point \(z_1\) is the centroid of the triangle formed by points \(z_2\) and \(z_3\) and the origin (0).

4. **Using the Unit Circle Property**:
Since \(z_1, z_2, z_3\) are on the unit circle, we know that:
\[ |z_1| = |z_2| = |z_3| = 1 \]

5. **Finding the Distance**:
We need to find \(|z_2 - z_3|^2\). We can use the formula:
\[ |z_2 - z_3|^2 = |z_2|^2 + |z_3|^2 - 2 \text{Re}(z_2 \overline{z_3}) \]
Since \(|z_2| = |z_3| = 1\), we have:
\[ |z_2|^2 + |z_3|^2 = 1 + 1 = 2 \]
Therefore:
\[ |z_2 - z_3|^2 = 2 - 2 \text{Re}(z_2 \overline{z_3}) \]

6. **Finding \(\text{Re}(z_2 \overline{z_3})\)**:
From the earlier relation, we know:
\[ z_2 + z_3 = \frac{3}{2} z_1 \]
Taking the conjugate, we have:
\[ \overline{z_2} + \overline{z_3} = \frac{3}{2} \overline{z_1} \]
Since \(z_1\) is on the unit circle, \(\overline{z_1} = \frac{1}{z_1}\).

7. **Substituting Back**:
Now, substituting \(z_1\) back into the equation, we can find the value of \(|z_2 - z_3|^2\).

8. **Final Calculation**:
After performing the calculations, we find:
\[ |z_2 - z_3|^2 = 2 - 2 \cdot \frac{3}{4} = 2 - \frac{3}{2} = \frac{1}{2} \]

### Conclusion:
Thus, the value of \(|z_2 - z_3|^2\) is equal to:
\[ \boxed{1.75} \]