The complex numbers `z_(1),z_(2),z_(3)` stisfying `(z_(2)-z_(3))=(1+i)(z_(1)-z_(3)).where i=sqrt(-1),` are vertices of a triangle which is

Options

A. equilateral
B. isosceles
C. right angled
D. scalene

Answer

To solve the problem, we need to analyze the given equation involving complex numbers and determine the type of triangle formed by the vertices represented by these complex numbers.

### Step-by-Step Solution:

1. **Given Equation**: We start with the equation:
\[ z_2 - z_3 = (1 + i)(z_1 - z_3) \]

2. **Rearranging the Equation**: We can express this in a more useful form:
\[ \frac{z_2 - z_3}{z_1 - z_3} = 1 + i \]

3. **Expressing in Polar Form**: To analyze the right side, we convert \(1 + i\) into polar form. We know:
\[ 1 + i = \sqrt{2} \left( \cos \frac{\pi}{4} + i \sin \frac{\pi}{4} \right) \]
Thus, we can write:
\[ \frac{z_2 - z_3}{z_1 - z_3} = \sqrt{2} e^{i \frac{\pi}{4}} \]

4. **Magnitude and Angle**: From the above, we can deduce:
- The magnitude:
\[ |z_2 - z_3| = \sqrt{2} |z_1 - z_3| \]
- The angle:
\[ \theta = \frac{\pi}{4} \]

5. **Setting Distances**: Let \( |z_1 - z_3| = x \). Then:
\[ |z_2 - z_3| = \sqrt{2} x \]

6. **Using the Cosine Rule**: To find the distance \( |z_2 - z_1| \), we apply the cosine rule in triangle \( z_1z_2z_3 \):
\[ |z_2 - z_1|^2 = |z_1 - z_3|^2 + |z_2 - z_3|^2 - 2 |z_1 - z_3| |z_2 - z_3| \cos \theta \]
Substituting the known values:
\[ |z_2 - z_1|^2 = x^2 + (\sqrt{2} x)^2 - 2 \cdot x \cdot \sqrt{2} x \cdot \cos \frac{\pi}{4} \]
\[ = x^2 + 2x^2 - 2 \cdot x \cdot \sqrt{2} x \cdot \frac{1}{\sqrt{2}} \]
\[ = 3x^2 - 2x^2 = x^2 \]
Thus, we find:
\[ |z_2 - z_1|^2 = x^2 \]

7. **Conclusion**: Since \( |z_2 - z_1| = |z_1 - z_3| \), we conclude that:
- The triangle is isosceles with two sides equal.
- Additionally, since the angle opposite the side \( |z_2 - z_1| \) is \( \frac{\pi}{2} \) (due to the angle \( \frac{\pi}{4} \) and the properties of the triangle), it is also a right triangle.

### Final Answer:
The triangle formed by the vertices \( z_1, z_2, z_3 \) is a **right-angled isosceles triangle**.

---