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.`

If the complex variables z_1,z_2 and origin form an equilateral triangle, prove that : z_1^2+z_2^2-z_1z_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).

bb"statement-1" " Let " z_(1),z_(2) " and " z_(3) be htree complex numbers, such that abs(3z_(1)+1)=abs(3z_(2)+1)=abs(3z_(3)+1) " and " 1+z_(1)+z_(2)+z_(3)=0, " then " z_(1),z_(2),z_(3) will represent vertices of an equilateral triangle on the complex plane. bb"statement-2" z_(1),z_(2),z_(3) represent vertices of an triangle, if z_(1)^(2)+z_(2)^(2)+z_(3)^(2)+z_(1)z_(2)+z_(2)z_(3)+z_(3)z_(1)=0

Let the complex numbers z_1,z_2,z_3 be the vertices of an equilateral triangle. Let z_0 be the circumcentre of the triangle. Then prove that z_1^2+z_2^2+z_3^2= 3z_0^2 .

Find the co-ordinates of the centroid of the triangle whose vertices are (x_(1),y_(1),z_(1)), (x_(2),y_(1),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)) .

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

The complex numbers z_1, z_2 and z_3 satisfying (z_1-z_3)/(z_2- z_3)= (1-i sqrt3)/2 are the vertices of triangle, which is :

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.