Prove that if |z1| = |z2| = |z3| = 1, and z1 + z2 + z3 = 0, then z1, z2, and z3 are vertices of an equilateral triangle inscribed in the unit circle with center at the origin. Graph it.

1. Given: \( |z_1| = |z_2| = |z_3| = 1 \), and \( z_1 + z_2 + z_3 = 0 \).

2. Express the numbers \( z_1, z_2, z_3 \) as points on the unit circle in the complex plane. Hence, \( z_1 = e^{i\alpha} \), \( z_2 = e^{i\beta} \), \( z_3 = e^{i\gamma} \) for some angles \( \alpha, \beta, \gamma \).

3. The condition \( z_1 + z_2 + z_3 = 0 \) implies that these points form an equilateral triangle.

4. Use symmetry and the condition \( z_1 + z_2 + z_3 = 0 \) to conclude:
z_1 = e^{i\alpha}, \, z_2 = e^{i(\alpha + \frac{2\pi}{3})}, \, z_3 = e^{i(\alpha + \frac{4\pi}{3})} .

5. Since the angles between them are \( \frac{2\pi}{3} \), they form an equilateral triangle.

6. The graph of these points would show an equilateral triangle inscribed in the unit circle centered at the origin.

7. Hence, proven: The vertices are indeed those of an equilateral triangle inscribed in the unit circle.

Answer: z_1 = e^{i\alpha}, \, z_2 = e^{i(\alpha + \frac{2\pi}{3})}, \, z_3 = e^{i(\alpha + \frac{4\pi}{3})}