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 the complex numbers z_(1) and z_(2) and the origin form an isosceles triangle with vertical angle (2pi)/(3) ,then show that z_(1)^(2)+z_(2)^(2)+z_(1)z_(2)=0.

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.

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

The points A,B,C represent the complex numbers z_1,z_2,z_3 respectively on a complex plane & the angle B & C of the triangle ABC are each equal to 1/2 (pi-alpha) . If (z_2-z_3)^2=lambda(z_3-z_1)(z_1-z_2) (sin^2) alpha/2 then determine lambda .

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.

A,B and C are the points respectively the complex numbers z_(1),z_(2) and z_(3) respectivley, on the complex plane and the circumcentre of /_\ABC lies at the origin. If the altitude of the triangle through the vertex. A meets the circumcircle again at P, prove that P represents the complex number (-(z_(2)z_(3))/(z_(1))) .

If z_(1),z_(2),z_(3) andz_(4) are the roots of the equation z^(4)=1, the value of sum_(i=1)^(4)(z_i)^(3) is

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

z_(1)=9+3i and z_(2)=-2-i then find z_(1)-z_(2)