The three vertices of a triangle are represented by the complex numbers ` 0, z_(1) and z_(2)` . If the triangle is equilateral, then

Options

A. ` z_(1)^(2) - z_(2)^(2) = z_(1) z_(2)`
B. ` z_(1)^(2) + z_(2)^(2) = z_(1) z_(2)`
C. ` z_(2)^(2) - z_(1)^(2) = z_(1) z_(2)`
D. `z_(1)^(2) + z_(2)^(2) + z_(1) z_(2) = 0 `

Answer

To determine the condition that must be satisfied for the vertices of a triangle represented by the complex numbers \( 0, z_1, \) and \( z_2 \) to form an equilateral triangle, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Geometry**:
- The vertices of the triangle are represented by the complex numbers \( 0, z_1, \) and \( z_2 \).
- For the triangle to be equilateral, the angles between the sides must be \( 60^\circ \) or \( \frac{\pi}{3} \) radians.

2. **Using the Argument of Complex Numbers**:
- The condition for the triangle to be equilateral can be expressed using the argument of the complex numbers.
- We can write the relationship between the complex numbers as:
\[ \frac{z_1}{z_2} = e^{i \frac{\pi}{3}} \]
- This implies:
\[ z_1 = z_2 e^{i \frac{\pi}{3}} \]

3. **Considering the Other Angle**:
- Similarly, we can consider the angle from \( z_2 \) to \( z_1 \):
\[ \frac{z_2}{z_1} = e^{-i \frac{\pi}{3}} \]
- This implies:
\[ z_2 = z_1 e^{-i \frac{\pi}{3}} \]

4. **Equating the Two Expressions**:
- From the first expression, we have:
\[ z_1 = z_2 e^{i \frac{\pi}{3}} \]
- From the second expression, substituting \( z_2 \) in terms of \( z_1 \):
\[ z_2 = z_1 e^{-i \frac{\pi}{3}} \]
- Substituting \( z_2 \) into the first equation gives:
\[ z_1 = (z_1 e^{-i \frac{\pi}{3}}) e^{i \frac{\pi}{3}} \]
- Simplifying this yields:
\[ z_1 = z_1 \]
- This confirms consistency.

5. **Finding the Relation**:
- We can also express the distances:
\[ |z_1| = |z_2| \quad \text{(since all sides are equal)} \]
- Therefore, we can write:
\[ z_1 - z_2 = z_1 e^{i \frac{\pi}{3}} - z_2 \]
- Rearranging gives:
\[ z_1 e^{i \frac{\pi}{3}} - z_2 = z_1 - z_2 \]
- This leads to the equation:
\[ z_1 z_2 - z_1^2 = z_2^2 \]
- Rearranging this gives:
\[ z_1^2 + z_2^2 = z_1 z_2 \]

### Final Condition:
Thus, the condition that must be satisfied for the triangle to be equilateral is:
\[ z_1^2 + z_2^2 = z_1 z_2 \]