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

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 :

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

If z_(1), z_(2), z_(3) be vertices of an equilateral triangle occurring in the anticlockwise sense then,

If z_(1), z_(2), z_(3) are three complex numbers in A.P., then they lie on

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), z_(3) satisfying (z_(1)-z_(3))/(z_(2)-z_(3))=(1-i sqrt(3))/(2) are the vertices of a triangle which is

Let z_(1) and z_(2) be two complex numbers such that |z_(1)+z_(2)|^(2)=|z_(1)|^(2)+|z_(2)|^(2) . Then,