If `z_1,z_2,z_3` are three points lying on the circle `|z|=2` then the minimum value of the expression `|z_1+z_2|^2+|z_2+z_3|^2+|z_3+z_1|^2=`

Explanation

To find the minimum value of the expression \( |z_1 + z_2|^2 + |z_2 + z_3|^2 + |z_3 + z_1|^2 \) where \( z_1, z_2, z_3 \) are points on the circle \( |z| = 2 \), we can follow these steps:

### Step 1: Rewrite the Expression
We start with the expression:
\[ |z_1 + z_2|^2 + |z_2 + z_3|^2 + |z_3 + z_1|^2 \]
Using the property \( |a|^2 = a \overline{a} \), we can expand each term:
\[ |z_1 + z_2|^2 = (z_1 + z_2)(\overline{z_1 + z_2}) = |z_1|^2 + |z_2|^2 + 2 \text{Re}(z_1 \overline{z_2}) \]
Thus, we have:
\[ |z_1 + z_2|^2 + |z_2 + z_3|^2 + |z_3 + z_1|^2 = (|z_1|^2 + |z_2|^2 + 2 \text{Re}(z_1 \overline{z_2})) + (|z_2|^2 + |z_3|^2 + 2 \text{Re}(z_2 \overline{z_3})) + (|z_3|^2 + |z_1|^2 + 2 \text{Re}(z_3 \overline{z_1})) \]

### Step 2: Substitute the Modulus Values
Since \( |z_1| = |z_2| = |z_3| = 2 \) (as they lie on the circle \( |z| = 2 \)), we have:
\[ |z_1|^2 = |z_2|^2 = |z_3|^2 = 4 \]
Thus, substituting these values gives:
\[ = 4 + 4 + 4 + 2 \text{Re}(z_1 \overline{z_2}) + 2 \text{Re}(z_2 \overline{z_3}) + 2 \text{Re}(z_3 \overline{z_1}) \]
This simplifies to:
\[ = 12 + 2 \left( \text{Re}(z_1 \overline{z_2}) + \text{Re}(z_2 \overline{z_3}) + \text{Re}(z_3 \overline{z_1}) \right) \]

### Step 3: Analyze the Real Part Terms
To minimize the expression, we need to minimize the term \( \text{Re}(z_1 \overline{z_2}) + \text{Re}(z_2 \overline{z_3}) + \text{Re}(z_3 \overline{z_1}) \). The maximum value of \( |z_1 \overline{z_2}| \) occurs when \( z_1 \) and \( z_2 \) are in the same direction, which is \( 4 \) (since \( |z_1| = |z_2| = 2 \)).

### Step 4: Use Geometric Interpretation
The points \( z_1, z_2, z_3 \) can be represented as vectors in the complex plane. The minimum value of the sum of the squares of the distances between these points occurs when they are symmetrically placed around the circle.

### Step 5: Calculate the Minimum Value
When \( z_1, z_2, z_3 \) are symmetrically placed (e.g., at \( 120^\circ \) apart), we can find:
\[ \text{Re}(z_1 \overline{z_2}) + \text{Re}(z_2 \overline{z_3}) + \text{Re}(z_3 \overline{z_1}) = -6 \]
Thus, substituting back into our expression gives:
\[ 12 + 2(-6) = 12 - 12 = 0 \]
However, since we are looking for the minimum value of the original expression, we find that the minimum value occurs when the vectors are aligned, leading to:
\[ = 12 + 0 = 12 \]

### Final Answer
The minimum value of the expression \( |z_1 + z_2|^2 + |z_2 + z_3|^2 + |z_3 + z_1|^2 \) is:
\[ \boxed{24} \]