Let `z_(1), z_(2), z_(3)` be three non-zero complex numbers such that `z_(1) bar(z)_(2) = z_(2) bar(z)_(3) = z_(3) bar(z)_(1)`, then `z_(1), z_(2), z_(3)`

Options

A. are vertices of an equilateral triangle
B. are vertices of an isosceles triangle
C. lie on a straight line
D. None of these

Answer

To solve the problem, we start with the given conditions for the complex numbers \( z_1, z_2, z_3 \):

Given:
\[ z_1 \overline{z_2} = z_2 \overline{z_3} = z_3 \overline{z_1} \]

### Step 1: Set the common value
Let \( k \) be the common value such that:
\[ z_1 \overline{z_2} = z_2 \overline{z_3} = z_3 \overline{z_1} = k \]

### Step 2: Express \( z_1, z_2, z_3 \) in terms of \( k \)
From the first equation:
\[ z_1 = \frac{k}{\overline{z_2}} \]
From the second equation:
\[ z_2 = \frac{k}{\overline{z_3}} \]
From the third equation:
\[ z_3 = \frac{k}{\overline{z_1}} \]

### Step 3: Take the modulus of the equations
Taking the modulus of the equations, we have:
\[ |z_1| |\overline{z_2}| = |k| \quad \Rightarrow \quad |z_1| |z_2| = |k| \]
\[ |z_2| |\overline{z_3}| = |k| \quad \Rightarrow \quad |z_2| |z_3| = |k| \]
\[ |z_3| |\overline{z_1}| = |k| \quad \Rightarrow \quad |z_3| |z_1| = |k| \]

### Step 4: Set the equations equal
From the above equations, we can set:
\[ |z_1| |z_2| = |z_2| |z_3| = |z_3| |z_1| \]

### Step 5: Cancel out \( |z_2| \)
Assuming \( |z_2| \neq 0 \), we can cancel \( |z_2| \) from the first two equations:
\[ |z_1| = |z_3| \]

### Step 6: Use the equality of moduli
From the second and third equations, we also have:
\[ |z_2| = |z_1| \]
\[ |z_3| = |z_2| \]

### Step 7: Conclude that all moduli are equal
Thus, we conclude:
\[ |z_1| = |z_2| = |z_3| \]

### Step 8: Establish the distances
Now, we can find the distances:
\[ |z_1 - z_2|^2 = |z_1|^2 + |z_2|^2 - 2 \text{Re}(z_1 \overline{z_2}) = 2|z_1|^2 - 2 \text{Re}(k) \]
\[ |z_2 - z_3|^2 = |z_2|^2 + |z_3|^2 - 2 \text{Re}(z_2 \overline{z_3}) = 2|z_2|^2 - 2 \text{Re}(k) \]
\[ |z_3 - z_1|^2 = |z_3|^2 + |z_1|^2 - 2 \text{Re}(z_3 \overline{z_1}) = 2|z_3|^2 - 2 \text{Re}(k) \]

### Step 9: Equate the distances
Since all moduli are equal, we find:
\[ |z_1 - z_2| = |z_2 - z_3| = |z_3 - z_1| \]

### Conclusion
Thus, the distances between the points are equal, which means that the points \( z_1, z_2, z_3 \) are the vertices of an equilateral triangle.

### Final Answer
Therefore, \( z_1, z_2, z_3 \) are the vertices of an equilateral triangle.

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