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|=2a n d(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_1^2+z_2^2+2z_1.z_2.costheta= 0 prove that the points represented by z_1, z_2 , and the origin form an isosceles triangle.

If (2z_1)/(3z_2) is purely imaginary then |(z_(1)-z_(2))/(z_(1)+z_(2))|

If a r g["z"_1("z"_3-"z"_2)]="a r g"["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_1, z_2, z_3 are complex numbers such that (2//z_1)=(1//z_2)+(1//z_3), then show that the points represented by z_1, z_2()_, z_3 lie on a circle passing through the origin.

If z_(1) = 1 - 3i, z_(2) = -4i and z_(3) = 5, show that (z_(1)z_(2)) z_(3) = z_(1) (z_(2)z_(3))

If z_(1) = 1 - 3i, z_(2) = -4i and z_(3) = 5, show that (z_(1) + z_(2)) + z_(3) = z_(1) + (z_(2) + z_(3))

If z_(1) = 3, z_(2) = -7i, and z_(3) = 5 + 4i, show that z_(1)(z_(2) + z_(3)) = z_(1) z_(2) + z_(1) z_(3)

If z_(1) = 3, z_(2) = -7i, and z_(3) = 5 + 4i, show that (z_(1) + z_(2)) z_(3) = z_(1) z_(3) + z_(2) z_(3)