Vertices A, B, C of an isosceles triangle ABC are represented by complex numbers z1, z2, z3 respectively. If $\angle C=90^{\circ}$ then which of the following is correct?

Question

Accepted Answer

To determine why Option 1 is the correct answer, we need to analyze the properties of an isosceles right triangle in the context of complex numbers.

### Explanation for Option 1:

1. **Isosceles Right Triangle Properties:**
   - In an isosceles right triangle, two sides are equal, and one angle is $90^\circ$.
   - Given that $\angle C = 90^\circ$, triangle ABC is an isosceles right triangle with $\angle C$ as the right angle.

2. **Complex Number Representation:**
   - Let $z_1$, $z_2$, and $z_3$ represent the vertices A, B, and C respectively.
   - The condition $\angle C = 90^\circ$ implies that the vectors representing the sides $z_1 - z_3$ and $z_2 - z_3$ are perpendicular.

3. **Mathematical Derivation:**
   - For vectors to be perpendicular, their dot product must be zero. In terms of complex numbers, this translates to:
     $$
     \text{Re}((z_1 - z_3) \overline{(z_2 - z_3)}) = 0
     $$
   - This condition can be rewritten using the property of complex numbers:
     $$
     (z_1 - z_2)^2 = 2(z_1 - z_3)(z_3 - z_2)
     $$
   - This equation represents the relationship between the sides of an isosceles right triangle in the complex plane, confirming that Option 1 is correct.

### Explanation for Why Other Options are Incorrect:

2. **Option 2:**
   - The equation $(z_1 - z_2)^2 = (z_1 - z_3)(z_3 - z_2)$ does not account for the factor of 2 that arises due to the isosceles nature of the triangle. This factor is crucial because it reflects the geometric properties of the triangle, specifically the equal lengths of the legs.

3. **Option 3:**
   - The equation $z_1^2 + z_2^2 + z_3^2 = z_1z_2z_3$ does not hold for an isosceles right triangle. This equation suggests a relationship that is not consistent with the geometric properties of such a triangle.

4. **Option 4:**
   - The equation $z_1^2 + z_2^2 + z_3^2 + z_1z_2z_3 = 0$ is not a valid representation of the relationships between the vertices of an isosceles right triangle. This equation implies a balance that does not exist in the context of the given triangle.

In conclusion, Option 1 correctly captures the relationship between the vertices of an isosceles right triangle in the complex plane, making it the correct answer. The other options fail to accurately represent the geometric and algebraic properties of such a triangle.