Let ` z_1, z_2 , z_3 in C ` such that ` |z_1 | = |z_2| = |z_3| = |z_1+ z_2+ z _3| = 4 `.
If ` |z_1 - z _2 | = | z _1 + z _ 3 | and z_2 ne z_3`, then values of ` |z_1 + z_2 |* |z_1 + z _ 3| ` is _____.

Explanation

To solve the problem step by step, we will use the properties of complex numbers and the given conditions.

### Step 1: Understanding the Given Conditions
We are given that:
- \( |z_1| = |z_2| = |z_3| = |z_1 + z_2 + z_3| = 4 \)
- \( |z_1 - z_2| = |z_1 + z_3| \)
- \( z_2 \neq z_3 \)

### Step 2: Geometric Interpretation
Since \( |z_1| = |z_2| = |z_3| = 4 \), we can interpret \( z_1, z_2, z_3 \) as points on a circle of radius 4 centered at the origin in the complex plane. The condition \( |z_1 + z_2 + z_3| = 4 \) indicates that the centroid of the triangle formed by these points is also on the circle.

### Step 3: Using the Condition \( |z_1 - z_2| = |z_1 + z_3| \)
This condition implies that the points \( z_1, z_2, z_3 \) are positioned such that the distance from \( z_1 \) to \( z_2 \) is equal to the distance from \( z_1 \) to \( z_3 \). This suggests that the angle \( \angle BAC \) (where \( A = z_1, B = z_2, C = z_3 \)) is \( 90^\circ \).

### Step 4: Applying the Geometric Properties
Since \( \angle BAC = 90^\circ \), we can use the Pythagorean theorem in the context of complex numbers:
\[ |z_1 - z_2|^2 + |z_1 - z_3|^2 = |z_2 - z_3|^2 \]

### Step 5: Finding \( |z_1 + z_2| \) and \( |z_1 + z_3| \)
Using the triangle inequality and the properties of complex numbers, we can express:
\[ |z_1 + z_2| = |z_1| + |z_2| = 4 + 4 = 8 \]
\[ |z_1 + z_3| = |z_1| + |z_3| = 4 + 4 = 8 \]

### Step 6: Calculate \( |z_1 + z_2| \cdot |z_1 + z_3| \)
Now we can find the product:
\[ |z_1 + z_2| \cdot |z_1 + z_3| = 8 \cdot 8 = 64 \]

### Final Answer
Thus, the value of \( |z_1 + z_2| \cdot |z_1 + z_3| \) is \( \boxed{64} \).