Neighboring Triples of Complex Numbers

Algebra Level 5

Two sets of complex numbers (z1,z2,z3,z4)C4(z_1, z_2, z_3, z_4) \in \mathbb{C^4} and (ω1,ω2,ω3)C3 (\omega_1 , \omega_2 , \omega_3 ) \in \mathbb{C^3} are called neighbors if all the following six numbers are purely real. ω1z1z2z1ω2z2z3z2ω3z3z1z3iω1z4z2z1iω2z4z3z2iω3z4z1z3\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 (z1,z2,z3,z4)C4(z_1, z_2, z_3, z_4) \in \mathbb{C^4} consisting of pairwise distinct complex numbers have a unique neighbor.

A quadruple (z1,z2,z3,z4)C4(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 ω1,ω2,ω3\omega_1, \omega_2, \omega_3 are equal).

A triple (z1,z2,z3)C3(z_1, z_2, z_3) \in \mathbb{C^3} is called attractive if for all xCx \in \mathbb{C}, the quadruple (z1,z2,z3,x)(z_1, z_2, z_3, x) is popular.

Let (z1,z2,z3)(z_1, z_2, z_3) be any attractive triple. Find (z2z1z3z1)\Im \left( \dfrac{z_2-z_1}{z_3-z_1}\right) .

Details and assumptions

  • i=1i = \sqrt{-1}

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


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