if the complex no `z_1 , z_2 and z_3 ` represents the vertices of an equilateral triangle such that `|z_1| = | z_2| = | z_3| ` then relation among `z_1 , z_2 and z_3 `

Explanation

Let us use the notion of rotation.
Let us rotate `R(z_(3)) " about " Q(z_(2))` in anticlockwise direction so that
` z_(1)-z_(2) = (|z_(1)-z_(2)|)/(|z_(3) -z_(2)|) (z_(3) -z_(2)) e^(pi/3) = ( PQ)/(QR) (z_(3) -z_(2)) e^(pi/3)`
` Rightarrow z_(1) -z_(2) =(z_(3) -z_(2)e^(pi/3))` ....(i)
Similarly by rotating `Q(z_(2)) " about " R(z_(3))` in closewise direction . we have ,` z_(1)-z_(2) = (z_(3) -z_(2)) e^((-ipi)/3)` ......(ii)
Mutilplying (i) & (ii) ,we get
`(z_(1)-z_(2))(z_(1)-z_(3))=(z_(3)-z_(2))e^((ipi)/3)(z_(2)-z_(3))e^((-ipi)/3)`
` Rightarrow z_(1)^(2) -z_(1)z_(3)-z_(1)z_(3) =z_(2)z_(3) -z_(3)^(2) -z_(2)^(2) +z_(2)z_(3)`
`Rightarrow z_(1)^(2) +z_(2)^(2) =z_(1)z_(2)+z_(2)z_(3)=z_(1)z_(3)`
which is the required condition for ` z_(1),z_(2),z_(3)` to represent vertices of an equilateral triangle.