Let `z_1, z_2a n dz_3` represent the vertices `A ,B ,a n dC` of the triangle `A B C ,` respectively, in the Argand plane, such that `|z_1|=|z_2|=|z_3|=5.` Prove that `z_1sin2A+z_2sin2B+z_3sin2C=0.`

Let A(z_1),B(z_2) and C(z_3) be the vertices of an equilateral triangle in the Argand plane such that |z_1|=|z_2|=|z_3|. Then (A) (z_2+z_3)/(2z_1-z_2-z_3) is purely real (B) (z_2-z_3)/(2z_1-z_2-z_3) is purely imaginary (C) |arg(z_1/z_2)|=2 arg((z_3-z_2)/(z_1-z_2))| (D) none of these

Let A(z_(1)),B(z_(2)),C(z_(3)) be the vertices of an equilateral triangle ABC in the Argand plane, then the number (z_(2)-z_(3))/(2z_(1)-z_(2)-z_(3)) , is

If z_(1), z_(2) and z_(3) are the vertices of a triangle in the argand plane such that |z_(1)-z_(2)|=|z_(1)-z_(3)| , then |arg((2z_(1)-z_(2)-z_(3))/(z_(3)-z_(2)))| is

If z_(1),z_(2) and z_(3) are the vertices of a right angledtriangle in Argand plane such that |z_(1)-z_(2)|=3,|z_(1)-z_(3)|=5 and z_(2) is the vertex with the right angle then

If z_1,z_2,z_3 be the vertices A,B,C respectively of triangle ABC such that |z_1|=|z_2|=|z_3| and |z_1+z_2|=|z_1-z_2| then C= (A) pi/2 (B) pi/3 (C) pi/6 (D) pi/4

if the complex no z_(1),z_(2) and z_(3) represents the vertices of an equilateral triangle such that |z_(1)|=|z_(2)|=|z_(3)| then relation among z_(1),z_(2) and z_(3)

If z_1, z_2, z_3 and z_(1)', z_(2)', z_(3)' represent the vertices of two similar triangles ABC and PQR, respectively then prove that |(bar(z_1))/(bar(z_2)- bar(z_1))|.|(z_2 - z_3)/(z_(3)')|+|(z_(2)')/(z_2-z_1)|.|(bar(z_3) - bar(z_1))/(bar(z_(3)'))| ge 1 .