A Problem in Complex Numbers!

Algebra Level 4

Let z1,z2,z3z_1,z_2,z_3 be 3 complex numbers lying on a circle whose centre is at the origin such that zi+zjzkz_i+z_{j}z_k (where i,j,k{1,2,3}i,j,k\in \{1,2,3\} and ijki \ne j \ne k) are real numbers, then find z1×z2×z3z_1\times z_2\times z_3.

Note

  • z1z2z3z_1\ne z_2\ne z_3. In other words, z1,z2,z3z_1,z_2,z_3 are all distinct points on the circle.

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