# Neighboring Triples of Complex Numbers

Algebra Level 5

Two sets of complex numbers $(z_1, z_2, z_3, z_4) \in \mathbb{C^4}$ and $(\omega_1 , \omega_2 , \omega_3 ) \in \mathbb{C^3}$ are called neighbors if all the following six numbers are purely real. $\dfrac{\omega_1-z_1}{z_2-z_1} \\ \\ \dfrac{\omega_2-z_2}{z_3-z_2} \\ \\ \dfrac{\omega_3-z_3}{z_1-z_3} \\ \\ i\dfrac{\omega_1-z_4}{z_2-z_1} \\ \\ i \dfrac{\omega_2-z_4}{z_3-z_2} \\ \\ i \dfrac{\omega_3-z_4}{z_1-z_3}$

It turns out that all quadruples $(z_1, z_2, z_3, z_4) \in \mathbb{C^4}$ consisting of pairwise distinct complex numbers have a unique neighbor.

A quadruple $(z_1, z_2, z_3, z_4) \in \mathbb{C^4}$ is called popular if the elements of its neighbor are equal (that is, the corresponding $\omega_1, \omega_2, \omega_3$ are equal).

A triple $(z_1, z_2, z_3) \in \mathbb{C^3}$ is called attractive if for all $x \in \mathbb{C}$, the quadruple $(z_1, z_2, z_3, x)$ is popular.

Let $(z_1, z_2, z_3)$ be any attractive triple. Find $\Im \left( \dfrac{z_2-z_1}{z_3-z_1}\right)$.

Details and assumptions

• $i = \sqrt{-1}$

• $\Im (z)$ denotes the imaginary part of $z$.

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