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.

If z, izand z+iz are the vertices of a triangle whose area is 2units, the value of |z| is

A,B and C are the points respectively the complex numbers z_(1),z_(2) and z_(3) respectivley, on the complex plane and the circumcentre of /_\ABC lies at the origin. If the altitude of the triangle through the vertex. A meets the circumcircle again at P, prove that P represents the complex number (-(z_(2)z_(3))/(z_(1))) .

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

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),z_(3) andz_(4) are the roots of the equation z^(4)=1, the value of sum_(i=1)^(4)(z_i)^(3) is

Suppose the points z_(1),z_(2),…,z_(n)(z_(i) ne 0) all lie on one side of a line drawn through the origin of the complex planes. Prove that the same is true of the points 1/z_(1),1/z_(2),...,1/z_(n) . Moreover, show that z_(1)+z_(2)+...+z_(n) ne 0 " and " 1/z_(1)+1/z_(2)+...+1/z_(n) ne 0

If z_(1),z_(2),z_(3),…………..,z_(n) are n nth roots of unity, then for k=1,2,,………,n

If |z_(1)|=|z_(2)|andamp(z_(1))+amp(z_(2))=0, then