Show that if z(1)z(2)+z(3)z(4)=0 and z(...

Show that if ` z_(1)z_(2)+z_(3)z_(4)=0 and z_(1)+z_(2)=0` ,then the complex numbers ` z_(1),z_(2),z_(3),z_(4)` are concyclic.

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We know that ` z_(1),z_(2),z_(3) and z_(4)` are concyclic if ` ,((z_(3)-z_(1))(z_(4)-z_(2)))/f((z_(3)-z_(2))(z_(4)-z_(1)))` is purely real.
According to the question, we have
`z_(1)+z_(2) =0 and z_(1)z_(2)+z_(3)z_(4)=0`
Now` ((z_(3)-z_(1))(z_(4)-z_(2)))/((z_(3)-z_(2))(z_(4)-z_(1)))=((z_(3)z_(4)+z_(1)z_(2))-(z_(2)z_(3)+z_(1)z_(4)))/((z_(3)z_(4)+z_(1)z_(2))-(z_(1)z_(3)+z_(2)z_(4)))`
`= (-(z_(2)z_(3)+z_(1)z_(4)))/((z_(1)z_(3)+z_(2)z_(4)))= (z_(2)(z_(3)-z_(4)))/(z_(2)(z_(4)-z_(3)))" as " z_(1)=z_(2)`
= -1 = purely real number
Consequently ` z_(1),z_(2),z_(3) and z_(4)` are concyclic points.

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