Prove that the complex numbers `z_(1) and z_(2)` and the origin form an isosceles triangle with vertical angle `(2pi)/(3),ifz_(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)).

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).

The complex numbers z_(1), z_(2), z_(3) are three vertices of a parallelogram taken in order then the fourth vertex is

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

If z_(1) and z_(2) are conjugate to each other , find the principal argument of (-z_(1)z_(2)) .

If the fourth roots of unity are z_(1), z_(2), z_(3), z_(4) then z_(1)^(2)+z_(2)^(2)+z_(3)^(2)+z_(4)^(2)=

Which of the following is correct for any two complex numbers z_(1) and z_(2) ?

If z_(1),z_(2),z_(3) represent the vertices of an equilateral triangle such that |z_(1)|=|z_(2)|=|z_(3)| , then

The three vertices of a triangle are represented by the complex numbers 0 , z_(1) and z_(2) . If the triangle is equilateral , then :

If z_(1),z_(2),z_(3) are the vertices of an equilateral triangle in the Argand plane, then (z_(1)^(2)+z_(2)^(2)+z_(3)^(3))=lambda(z_(1)z_(2)+z_(2)z_(3)+z_(3)z_(1)) holds true when :