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Find the condition in order that z(1),z...

Find the condition in order that ` z_(1),z_(2),z_(3)` are vertices of an.isosceles triangle right angled at ` z_(2)`.

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Let ` P(z_(1)) Q(z_(2)) and R(z_(3))` the vertices of an isosceles tirangle PQR right angle of Q. Let us rotate `Q(z_(2))` about P in anticlockwise direction,we have. `z_(3)-z_(1) (|z_(3)-z_(1)|)/(|z_(2)-z_(1)|) (z_(2)-z_(1))e^(pi/4)`
` = (PR)/(PQ)(z_(2)-z_(1))e^((ipi)/4)=sqrt2(z_(2)-z_(1))e^((ipi)/4)`
Also by rotating `Q(z_(2)) " about" R(z_(3))` in clockwise direction.
`z_(1)-z_(3)= (|z_(1)-z_(3)|)/(|z_(2)-z_(3)|) (z_(2)-z_(3))e^((-ipi)/4)= (PR)/(QR) (z_(2)-z_(3))e^((-ipi)/4) =sqrt2(z_(2)-z_(3))e^((-ipi)/4).....(ii)`
Multiplying (i) & (ii) we get
`(z_(3)-z_(1))(z_(1)-z_(3))=2(z_(2)-z_(2)) (z_(2)-z_(3))`
Which is the required condition .
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