If the complex numbers `z_(1) and z_(2)` and the origin form an isosceles triangle with vertical angle `(2pi)/(3)`,then show that `z_(1)^(2)+z_(2)^(2)+z_(1)z_(2)=0.`

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)).

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.

Complex numbers z_(1),z_(2),z_(3) are the vertices of A,B,C respectively of an equilteral triangle. Show that z_(1)^(2)+z_(2)^(2)+z_(3)^(2)=z_(1)z_(2)+z_(2)z_(3)+z_(3)z_(1).

z_(1) and z_(2) are two complex number such that |z_(1)|=|z_(2)| and arg (z_(1))+arg(z_(2))=pi , then show that z_(1)=-barz_(2)

Find the circumcentre of the triangle whose vertices are given by the complex numbers z_(1),z_(2) and z_(3) .

If the complex numbers z_(1) and z_(2) arg (z_(1))- arg(z_(2))=0 then showt aht |z_(1)-z_(2)|=|z_(1)|-|z_(2)| .

Find the coordinates of the centroid of the triangle whose vertices are (x_(1),y_(1),z_(1)),(x_(2),y_(2),z_(2))and(x_(3),y_(3),z_(3)) .

For any two complex numbers z_(1) and z_(2) , prove that Re ( z_(1)z_(2)) = Re z_(1) Re z_(2)- 1mz_(1) Imz_(2)

z_(1)=3+I and z_(2)=-2+6i then find z_(1)z_(2)

If the points represented by complex numbers z_(1)=a+ib, z_(2)=c+id " and " z_(1)-z_(2) are collinear, where i=sqrt(-1) , then