If `A(z_(1)),B(z_(2)), C(z_(3))` are the vertices of an equilateral triangle ABC, then arg `(2z_(1)-z_(2)-z_(3))/(z_(3)_z_(2))=`

Options

A. `pi/4`
B. `pi/2`
C. `pi/3`
D. `pi/6`

Explanation

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

### Step-by-step Solution:

1. **Understanding the Geometry**:
- Let \(z_1\), \(z_2\), and \(z_3\) represent the complex numbers corresponding to the vertices \(A\), \(B\), and \(C\) of the equilateral triangle. The angles between the sides of the triangle are \(60^\circ\) or \(\frac{\pi}{3}\) radians.

2. **Rearranging the Expression**:
- We need to simplify the expression \(\frac{2z_1 - z_2 - z_3}{z_3 - z_2}\).

3. **Expressing \(z_1\) in terms of \(z_2\) and \(z_3\)**:
- We can express \(z_1\) as a rotation of \(z_2\) and \(z_3\). Since \(z_1\) is at an angle of \(\frac{\pi}{3}\) from the line segment \(z_2z_3\), we can write:
\[ z_1 - z_2 = |z_1 - z_2| e^{i\frac{\pi}{3}} \]
\[ z_1 - z_3 = |z_1 - z_3| e^{-i\frac{\pi}{3}} \]

4. **Substituting into the Expression**:
- Substitute \(z_1 - z_2\) and \(z_1 - z_3\) into the expression:
\[ 2z_1 - z_2 - z_3 = 2(z_2 + |z_1 - z_2| e^{i\frac{\pi}{3}}) - z_2 - (z_2 + |z_1 - z_3| e^{-i\frac{\pi}{3}}) \]
- This simplifies to:
\[ |z_1 - z_2| e^{i\frac{\pi}{3}} - |z_1 - z_3| e^{-i\frac{\pi}{3}} \]

5. **Calculating the Argument**:
- The expression \(\frac{2z_1 - z_2 - z_3}{z_3 - z_2}\) can be simplified further. We can factor out \(z_3 - z_2\) from the numerator:
\[ \frac{(z_3 - z_2)(\text{some complex number})}{z_3 - z_2} \]
- This leads to:
\[ \text{arg}\left(2z_1 - z_2 - z_3\right) - \text{arg}(z_3 - z_2) \]

6. **Final Argument Calculation**:
- The argument of \(z_3 - z_2\) is simply the angle of the line segment from \(z_2\) to \(z_3\).
- The argument of \(2z_1 - z_2 - z_3\) can be calculated based on the angles involved. Since \(z_1\) is at \(60^\circ\) from both \(z_2\) and \(z_3\), we find that:
\[ \text{arg}(2z_1 - z_2 - z_3) = \frac{\pi}{2} \]

7. **Conclusion**:
- Therefore, the final result is:
\[ \text{arg}\left(\frac{2z_1 - z_2 - z_3}{z_3 - z_2}\right) = \frac{\pi}{2} \]