Complex numbers `z_(1),z_(2)andz_(3)` are the vertices A,B,C respectivelt of an 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))`