Show that the triangle whose vertices are `z_(1)z_(2)z_(3)andz_(1)'z_(2)'z_(3)'` are directly similar , if `|{:(z_(1),z'_(1),1),(z_(2),z'_(2),1),(z_(3),z'_(3),1):}|=0`

Show that the triangle whose vertices are z_1,z_2,z_3 and barz_1, barz_2 and barz_3 and z_1\',z_2\',z_3\' are similar if |(z_1,z_1\',1),(z_2,z_2\', 1),(z_3, z_3\' 1)|=0 or z_1(z_2\'-z_3\')+z_2(z_3\'-z_1\')+(z_1\'-z_2\')=0

If two triangles whose vertices are respectively the complex numbers z_(1),z_(2),z_(3) and a_(1),a_(2),a_(3) are similar, then the determinant. |{:(z_(1),a_(1),1),(z_(2),a_(2),1),(z_(3),a_(3),1):}| is equal to

Let z_(0) be the circumcenter of an equilateral triangle whose affixes are z_(1),z_(2),z_(3) . Statement-1 : z_(1)^(2)+z_(2)^(2)+z_(3)^(2)=3z_(0)^(2) Statement-2: z_(1)^(2)+z_(2)^(2)+z_(3)^(2)=2(z_(1)z_(2)+z_(2)z_(3)+z_(3)z_(1))

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.

, a point 'z' is equidistant from three distinct points z_(1),z_(2) and z_(3) in the Argand plane. If z,z_(1) and z_(2) are collinear, then arg (z(z_(3)-z_(1))/(z_(3)-z_(2))). Will be (z_(1),z_(2),z_(3)) are in anticlockwise sense).

If |z_(1)|=2,|z_(2)|=3,|z_(3)|=4and|z_(1)+z_(2)+z_(3)|=5. "then"|4z_(2)z_(3)+9z_(3)z_(1)+16z_(1)z_(2)| is

If |z_(1)|=|z_(2)|=|z_(3)|=1 and z_(1)+z_(2)+z_(3)=sqrt(2)+i , then the complex number z_(2)barz_(3)+z_(3)barz_(1)+z_(1)barz_(2) , is

If az_(1)+bz_(2)+cz_(3)=0 for complex numbers z_(1),z_(2),z_(3) and real numbers a,b,c then z_(1),z_(2),z_(3) lie on a

Prove that traingle by complex numbers z_(1),z_(2) and z_(3) is equilateral if |z_(1)|=|z_(2)| = |z_(3)| and z_(1) + z_(2) + z_(3)=0

If the tangents at z_(1) , z_(2) on the circle |z-z_(0)|=r intersect at z_(3) , then ((z_(3)-z_(1))(z_(0)-z_(2)))/((z_(0)-z_(1))(z_(3)-z_(2))) equals