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 \(A\), \(B\), and \(C\) be the vertices of the equilateral triangle in the complex plane, represented by the complex numbers \(z_1\), \(z_2\), and \(z_3\) respectively.

2. **Finding the Midpoint**:
- The midpoint \(D\) of segment \(BC\) can be calculated as:
\[ D = \frac{z_2 + z_3}{2} \]

3. **Using the Property of Equilateral Triangles**:
- In an equilateral triangle, the line segment from a vertex to the midpoint of the opposite side is perpendicular to that side. Therefore, the line \(AD\) is perpendicular to line \(BC\).

4. **Setting Up the Argument**:
- Since \(AD\) is perpendicular to \(BC\), we can express this relationship in terms of arguments:
\[ \text{arg}(z_1 - D) - \text{arg}(z_3 - z_2) = \frac{\pi}{2} \]
- This implies:
\[ \text{arg}(z_1 - \frac{z_2 + z_3}{2}) = \text{arg}(z_3 - z_2) + \frac{\pi}{2} \]

5. **Rearranging the Expression**:
- We can rewrite \(z_1\) in terms of \(z_2\) and \(z_3\):
\[ z_1 = D + r \cdot e^{i\theta} \]
- Here, \(r\) is the distance from \(D\) to \(A\) and \(\theta\) is the angle corresponding to the position of \(A\) relative to \(D\).

6. **Calculating the Argument**:
- Now substituting \(D\) back into the expression:
\[ \text{arg}\left(2z_1 - z_2 - z_3\right) = \text{arg}\left(2\left(\frac{z_2 + z_3}{2} + r \cdot e^{i\theta}\right) - z_2 - z_3\right) \]
- Simplifying gives:
\[ \text{arg}(2r \cdot e^{i\theta}) = \text{arg}(z_3 - z_2) + \frac{\pi}{2} \]

7. **Final Result**:
- Therefore, we conclude that:
\[ \text{arg}\left(\frac{2z_1 - z_2 - z_3}{z_3 - z_2}\right) = \frac{\pi}{2} \]

### Final Answer:
\[ \text{arg}\left(\frac{2z_1 - z_2 - z_3}{z_3 - z_2}\right) = \frac{\pi}{2} \]