If `z_(1), z_(2) and z_(3)` are the vertices of a triangle in the argand plane such that `|z_(1)-z_(2)|=|z_(1)-z_(3)|`, then `|arg((2z_(1)-z_(2)-z_(3))/(z_(3)-z_(2)))|` is

Options

A. `(pi)/(3)`
B. 0
C. `(pi)/(2)`
D. `(pi)/(6)`

Answer

To solve the problem, we need to analyze the condition given and derive the required expression step by step.

### Step 1: Understand the Condition
Given that \( |z_1 - z_2| = |z_1 - z_3| \), this implies that the point \( z_1 \) is equidistant from \( z_2 \) and \( z_3 \). Therefore, the triangle formed by the points \( z_1, z_2, z_3 \) is isosceles with \( z_1 \) at the apex.

**Hint:** Visualize the triangle in the Argand plane to understand the geometric implications of the given condition.

### Step 2: Use the Rotation Concept
We can express the relationship between the complex numbers using the rotation concept. We can write:
\[ \frac{z_1 - z_2}{z_3 - z_2} = r e^{i\theta} \]
where \( r = \frac{|z_1 - z_2|}{|z_3 - z_2|} = 1 \), since \( |z_1 - z_2| = |z_1 - z_3| \).

**Hint:** Recall that the argument of a complex number can be interpreted as the angle it makes with the positive real axis.

### Step 3: Express the Required Argument
We need to evaluate:
\[ \left| \arg\left( \frac{2z_1 - z_2 - z_3}{z_3 - z_2} \right) \right| \]
We can rewrite \( 2z_1 - z_2 - z_3 \) as:
\[ 2z_1 - z_2 - z_3 = (z_1 - z_2) + (z_1 - z_3) \]
Using our previous result, we can substitute:
\[ z_1 - z_2 = re^{i\theta}, \quad z_1 - z_3 = re^{-i\theta} \]

**Hint:** Use the properties of complex numbers to simplify the expression.

### Step 4: Simplify the Expression
Now substituting back, we have:
\[ 2z_1 - z_2 - z_3 = re^{i\theta} + re^{-i\theta} = r \left( e^{i\theta} + e^{-i\theta} \right) = 2r \cos(\theta) \]
Thus:
\[ \frac{2z_1 - z_2 - z_3}{z_3 - z_2} = \frac{2r \cos(\theta)}{z_3 - z_2} \]

**Hint:** Remember that \( z_3 - z_2 \) is a complex number whose argument contributes to the overall argument of the fraction.

### Step 5: Determine the Argument
Since \( z_3 - z_2 \) is a complex number, the argument of the entire expression can be simplified:
\[ \arg\left( \frac{2z_1 - z_2 - z_3}{z_3 - z_2} \right) = \arg(2r \cos(\theta)) + \arg(z_3 - z_2) \]
The term \( 2r \cos(\theta) \) is real, and thus its argument is \( 0 \) or \( \pi \) depending on the sign.

**Hint:** Consider the nature of the angles involved and how they relate to the isosceles triangle.

### Step 6: Conclusion
Since the argument of a real number is either \( 0 \) or \( \pi \), we conclude that:
\[ \left| \arg\left( \frac{2z_1 - z_2 - z_3}{z_3 - z_2} \right) \right| = \frac{\pi}{2} \]

Thus, the final answer is:
\[ \boxed{\frac{\pi}{2}} \]