In Fig. 42, a point 'z' is equidistant from three distinct points `z_(1),z_(2)` and `z_(3)` in the Argand plane. If `z,z_(1)` and `z_(2)` are collinear, then arg`((z_(3)-z_(1))/(z_(3)-z_(2)))`. Will be `(z_(1),z_(2),z_(3))` are in anticlockwise sense).

Options

A. `pi/2`
B. `-pi/2`
C. `pi/6`
D. `(2pi)/3`

Explanation

To solve the problem, we need to analyze the geometric configuration of the points in the Argand plane and use properties of complex numbers and their arguments.

### Step-by-Step Solution:

1. **Understanding the Configuration**:
- Let \( z_1, z_2, z_3 \) be the three distinct points in the Argand plane.
- The point \( z \) is equidistant from \( z_1, z_2, z_3 \).
- Since \( z_1, z_2, z_3 \) are collinear and \( z \) is equidistant from them, \( z \) lies on the perpendicular bisector of the segment joining \( z_1 \) and \( z_2 \).

2. **Using the Property of Equidistance**:
- Since \( z \) is equidistant from \( z_1 \) and \( z_2 \), we have:
\[ |z - z_1| = |z - z_2| \]
- This implies that the point \( z \) lies on the perpendicular bisector of the line segment joining \( z_1 \) and \( z_2 \).

3. **Identifying the Angles**:
- Let \( P \) be the point \( z \).
- The angles formed at point \( P \) with respect to points \( z_1 \) and \( z_2 \) can be denoted as:
- \( \angle Pz_1z_3 \)
- \( \angle Pz_2z_3 \)
- Since \( z_1, z_2, z_3 \) are in anticlockwise order, the angle \( \angle z_2z_3z_1 \) is \( 90^\circ \) (or \( \frac{\pi}{2} \) radians).

4. **Calculating the Argument**:
- We need to find:
\[ \text{arg}\left(\frac{z_3 - z_1}{z_3 - z_2}\right) \]
- From the geometric configuration, we know that:
\[ \text{arg}(z_3 - z_1) - \text{arg}(z_3 - z_2) = \frac{\pi}{2} \]
- Rearranging gives:
\[ \text{arg}(z_3 - z_1) = \text{arg}(z_3 - z_2) + \frac{\pi}{2} \]

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