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If z(1),z(2)andz(3) are the vertices of...

If `z_(1),z_(2)andz_(3)` are the vertices of an equilasteral triangle with `z_(0)` as its circumcentre , then changing origin to `z^(0)` ,show that `z_(1)^(2)+z_(2)^(2)+z_(3)^(2)=0,` where`z_(1),z_(2),z_(3),` are new complex numbers of the vertices.

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Let `A(z_(a)),B(z_(b))` and `C(z_(c))` be the vertices of the trinagle.

Considering the rotation about points `A_(1)` and `B_(1)`, respectively, we get
`(z_(c)-z_(1))/(0-z_(1)) =(|z_(c) - z_(1)|)/(r) e^(-pi//2) and (z_(c) - z_(2))/(0-z_(2)) = (|z_(c) - z_(2)|)/(r)e^(ipi//2)`
`((z_(c)-z_(1))z_(2))/(z_(1)(z_(c)-z_(2)))=e^(-ipi) = -1`
`rArr (z_(c)- z_(1))z_(2) = - z_(1)(z_(c)-z_(2))`
or `z_(c)= (2z_(1)z_(2))/((z_(1) +z_(2))`
Similarly,
`z_(a)=(2z_(2)z_(3))/((z_(2)+z_(3))` and `z_(b)=(2z_(1)z_(3))/((z_(1)+z_(3))`
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