If arg `(z-z_1)/(z_2-z_1)` = 0 for three distinct complex numbers `z,z_1,z_2` then the three points are

Options

A. concyclin
B. vertices of an equilateral triangle
C. collinear
D. vertics of a right-angled triangle

Answer

To solve the problem, we need to analyze the condition given by the argument of the complex numbers. The condition states that:

\[ \text{arg}\left(\frac{z - z_1}{z_2 - z_1}\right) = 0 \]

### Step 1: Understanding the Argument Condition
The argument of a complex number is zero when the complex number is a positive real number. Therefore, the expression \(\frac{z - z_1}{z_2 - z_1}\) must be a positive real number.

### Step 2: Setting Up the Condition
Since \(\frac{z - z_1}{z_2 - z_1}\) is a positive real number, we can express this condition as:

\[ z - z_1 = k(z_2 - z_1) \]

for some \(k > 0\). This implies that \(z\) can be expressed in terms of \(z_1\) and \(z_2\).

### Step 3: Rearranging the Equation
Rearranging the equation gives us:

\[ z = z_1 + k(z_2 - z_1) \]

### Step 4: Interpretation of the Result
The equation \(z = z_1 + k(z_2 - z_1)\) indicates that \(z\) lies on the line segment that connects \(z_1\) and \(z_2\). Since \(k\) is a positive real number, \(z\) can take any point along the line extending from \(z_1\) through \(z_2\).

### Conclusion
Thus, the three points \(z\), \(z_1\), and \(z_2\) are collinear. This means that they lie on the same straight line in the complex plane.

### Final Answer
The three points \(z\), \(z_1\), and \(z_2\) are collinear.

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