Complex numbers of `z_(1),z_(2),z_(3)` are the vertices A, B, C respectively, of on isosceles right-angled triangle with right angle at C. show that `(z_(1) - z_(2))^(2)` = `2 (z_(1) - z_(3)) (z_(3)- z_(2))`

Explanation

Applying rotation about point B,
`(z_(1)-z_(2))/(z_(3)-z_(2))=sqrt(2)e^(ipi//4)" "(1)`
Applying rotation about point A,
`(z_(2)-z_(1))/(z_(3)-z_(1))=sqrt(2)e^(-ipi//4)`
Multiplying (1) and (2), we get
`((z_(1)-z_(2))(z_(2)-z_(1)))/((z_(3)-z_(2))(z_(3)-z_(1)))=2`
`(z_(1)-z_(2))^(2)=-2(z_(3)-z_(2))(z_(3)-z_(1))`
`=2(z_(1)-z_(3))(z_(3)-z_(2))`