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`.

If z is any complex number, prove that : |z|^2= |z^2| .

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),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).

If z_1 and z_2 (ne 0) are two complex numbers, prove that : |z_1 z_2|= |z_1||z_2| .

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 .

If z_1 and z_2 (ne 0) are two complex numbers, prove that : |z_1/z_2|= (|z_1|)/ (|z_2|), z_2 ne 0 .

If two complex numbers z_1,z_2 are such that |z_1|= |z_2| , is it then necessary that z_1 = z_2 ?

If the complex number z_1 and z_2 be such that arg(z_1)-arg(z_2)=0 , then show that |z_1-z_2|=|z_1|-|z_2| .

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