If `z_0` is the circumcenter of an equilateral triangle with vertices `z_1, z_2, z_3` then `z_1^2+z_2^2+z_3^2` is equal to

If z_1, z_2 and z_3 , are the vertices of an equilateral triangle ABC such that |z_1 -i = |z_2 -i| = |z_3 -i| .then |z_1 +z_2+ z_3| equals:

If the complex numbers z_(1), z_(2)" and "z_(3) denote the vertices of an isoceles triangle, right angled at z_(1), " then "(z_(1)-z_(2))^(2)+(z_(1)-z_(3))^(2) is equal to

Suppose z_(1),z_(2)andz_(3) are the vertices of an equilateral triangle inscribed in the circle |z| = 2. If z_(1)=1+isqrt3 then find z_(2)andz_(3) .

Let z_1, z_2, z_3 be the three nonzero complex numbers such that z_2!=1,a=|z_1|,b=|z_2|a n d c=|z_3|dot Let |(a, b, c), (b, c, a), (c,a,b) |=0 a r g(z_3)/(z_2) equal to (a) arg((z_3-z_1)/(z_2-z_1))^2 (b) orthocentre of triangle formed by z_1, z_2, z_3, i sz_1+z_2+z_3 (c)if triangle formed by z_1, z_2, z_3 is equilateral, then its area is (3sqrt(3))/2|z_1|^2 (d) if triangle formed by z_1, z_2, z_3 is equilateral, then z_1+z_2+z_3=0

Let vertices of an acute-angled triangle are A(z_1),B(z_2),a n dC(z_3)dot If the origin O is he orthocentre of the triangle, then prove that z_1( bar z )_2+( bar z )_1z_2=z_2( bar z )_3+( bar z )_2z_3=z_3( bar z )_1+( bar z )_(3)z_1

Complex numbers z_1 , z_2 , z_3 are the vertices A, B, C respectively of an isosceles right angled trianglewith right angle at C and (z_1- z_2)^2 = k(z_1 - z_3) (z_3 -z_2) , then find k.

If the points A(z),B(-z),a n dC(1-z) are the vertices of an equilateral triangle A B C , then (a)sum of possible z is 1/2 (b)sum of possible z is 1 (c)product of possible z is 1/4 (d)product of possible z is 1/2

z_1a n dz_2 lie on a circle with center at the origin. The point of intersection z_3 of the tangents at z_1a n dz_2 is given by a. 1/2(z_1+( z )_2) b. (2z_1z_2)/(z_1+z_2) c. 1/2(1/(z_1)+1/(z_2)) d. (z_1+z_2)/(( z )_1( z )_2)

On the Argand plane z_1, z_2a n dz_3 are respectively, the vertices of an isosceles triangle A B C with A C=B C and equal angles are thetadot If z_4 is the incenter of the triangle, then prove that (z_2-z_1)(z_3-z_1)=(1+sectheta)(z_4-z_1)^2dot