Let `A(z_(1)),B(z_(2)),C(z_(3))` be the vertices of an equilateral triangle ABC in the Argand plane, then the number `(z_(2)-z_(3))/(2z_(1)-z_(2)-z_(3))`, is

Options

A. purely real
B. purely imaginary
C. a complex number with non-zero and imaginary parts
D. none of these

Answer

To solve the problem, we need to evaluate the expression \(\frac{z_2 - z_3}{2z_1 - z_2 - z_3}\) where \(A(z_1), B(z_2), C(z_3)\) are the vertices of an equilateral triangle in the Argand plane.

### Step-by-Step Solution:

1. **Understanding the Geometry**:
Since \(A\), \(B\), and \(C\) are vertices of an equilateral triangle, the angles between each pair of vertices are \(60^\circ\) (or \(\frac{\pi}{3}\) radians).

2. **Expressing the Difference**:
We start with the expression:
\[ \frac{z_2 - z_3}{2z_1 - z_2 - z_3} \]

3. **Rewriting the Denominator**:
The denominator \(2z_1 - z_2 - z_3\) can be rewritten as:
\[ 2z_1 - z_2 - z_3 = z_1 - z_2 + z_1 - z_3 \]
This can also be expressed in terms of the differences:
\[ = (z_1 - z_2) + (z_1 - z_3) \]

4. **Using Rotations**:
In an equilateral triangle, rotating a point by \(60^\circ\) (or \(\frac{\pi}{3}\) radians) gives us:
\[ z_1 - z_2 = (z_2 - z_3)e^{i\frac{\pi}{3}} \]
and
\[ z_1 - z_3 = (z_2 - z_3)e^{-i\frac{\pi}{3}} \]

5. **Substituting Back**:
Substituting these into the denominator:
\[ 2z_1 - z_2 - z_3 = (z_2 - z_3)e^{i\frac{\pi}{3}} + (z_2 - z_3)e^{-i\frac{\pi}{3}} \]

6. **Factoring Out**:
Factoring out \(z_2 - z_3\):
\[ = (z_2 - z_3) \left(e^{i\frac{\pi}{3}} + e^{-i\frac{\pi}{3}}\right) \]

7. **Using Euler's Formula**:
Using Euler's formula, we know:
\[ e^{i\frac{\pi}{3}} + e^{-i\frac{\pi}{3}} = 2\cos\left(\frac{\pi}{3}\right) = 2 \cdot \frac{1}{2} = 1 \]

8. **Final Expression**:
Therefore, the denominator simplifies to:
\[ 2z_1 - z_2 - z_3 = (z_2 - z_3) \cdot 1 = z_2 - z_3 \]

9. **Simplifying the Original Expression**:
Now substituting back into the original expression:
\[ \frac{z_2 - z_3}{z_2 - z_3} = 1 \]

### Conclusion:
The value of \(\frac{z_2 - z_3}{2z_1 - z_2 - z_3}\) is \(1\).