If `((3-z_1)/(2-z_1))((2-z_2)/(3-z_2))=k(k >0)` , then prove that points `A(z_1),B(z_2),C(3),a n dD(2)` (taken in clockwise sense) are concyclic.

If z_(1)=iz_(2) and z_(1)-z_(3)=i(z_(3)-z_(2)), then prove that |z_(3)|=sqrt(2)|z_(1)|

If arg[z_(3)-z_(2))]=arg[z_(3)(z_(2)-z_(1))], then find prove that O,z_(1),z_(2),z_(3) are concyclic, where O is the origin.

If |z|=2and(z_(1)-z_(3))/(z_(2)-z_(3))=(z-2)/(z+2), then prove that z_(1),z_(2),z_(3) are vertices of a right angled triangle.

If z_(1),z_(2),z_(3) are complex numbers such that (2)/(z_(1))=(1)/(z_(2))+(1)/(z_(3)) , then the points z_(1),z_(2),z_(3) and origin are

If z_(1),z_(2),z_(3),z_(4) are the affixes of four point in the Argand plane,z is the affix of a point such that |z-z_(1)|=|z-z_(2)|=|z-z_(3)|=|z-z_(4)| ,then prove that z_(1),z_(2),z_(3),z_(4) are concyclic.

Show that if z_(1)z_(2)+z_(3)z_(4)=0 and z_(1)+z_(2)=0 ,then the complex numbers z_(1),z_(2),z_(3),z_(4) are concyclic.

Let z_1 , z _2 and z_3 be three complex numbers such that z_1 + z_2+ z_3 = z_1z_2 + z_2z_3 + z_1 z_3 = z_1 z_2z_3 = 1 . Then the area of triangle formed by points A(z_1 ), B(z_2) and C(z_3) in complex plane is _______.

If |z_(1)|=|z_(2)|=|z_(3)| and z_(1)+z_(2)+z_(3)=0 , then z_(1),z_(2),z_(3) are vertices of

if A(z_1),B(z_2),C(z_3),D(z_4) lies on |z|=4 (taken in order) , where z_1+z_2+z_3+z_4=0 then :