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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