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.

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.

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 |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 arg(z_(2)-z_(1))+arg(z_(3)-z_(1))=2arg(z-z_(1)) where z,z_(1),z_(2) and z_(3) ,are complex numbers, then the locus of z is

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.

If z_(1),z_(2),z_(3) satisfy the system of equations given by |z_(1)|=|z_(2)|=|z+3|=1,z_(1)+z_(2)+z_(3)=1 and z_(1)z_(2)z_(3)=1 such that lm(z_(1))

If 2z_(1)-3z_(2)+z_(3)=0, then z_(1),z_(2) and z_(3) are represented by

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 P and Q are represented by the complex numbers z_1 and z_2 such that |(1)/(z_2) + (1)/(z_1)|= |(1)/(z_2) - (1)/(z_1)| , then the circumstance of Delta OPQ (where O is the origin) is