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).

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.

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 .

The points A(x,y), B(y, z) and C(z,x) represents the vertices of a right angled triangle, if

If z_1,z_2 in C , show that (z_1+ z_2)^2= z_1^2+2 z_1 z_2+ z_2^2 .

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.

If z_(1),z_(2)andz_(3) are the affixes of the vertices of a triangle having its circumcentre at the origin. If zis the affix of its orthocentre, prove that Z_(1)+Z_(2)+Z_(3)-Z=0.

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 :

The points z_(1),z_(2),z_(3),z_(4) in the complex plane are the vertices of a parallelogram taken in order if and only if

The complex numbers z_(1),z_(2),z_(3) stisfying (z_(2)-z_(3))=(1+i)(z_(1)-z_(3)).where i=sqrt(-1), are vertices of a triangle which is