IF the vertices of a triange ABC are respresented by `z_1,z_2 and z_3` respectively, show that its circumcentre is `(z_1sin2A+z_2sin2B+z_3sin2C)/(sin2A+sin2B+sin2C)`

Let z_(1),z_(2) and z_(3) represent the vertices A,B, and C of the triangle ABC , respectively,in the Argand plane,such that |z_(1)|=|z_(2)|=|z_(3)|=5. Prove that z_(1)sin2A+z_(2)sin2B+z_(3)sin2C=0

If z_(1) , z_(2) and z_(3) are the vertices of DeltaABC , which is not right angled triangle taken in anti-clock wise direction and z_(0) is the circumcentre, then ((z_(0)-z_(1))/(z_(0)-z_(2)))(sin2A)/(sin2B)+((z_(0)-z_(3))/(z_(0)-z_(2)))(sin2C)/(sin2B) is equal to

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

Show that sin (x-y) + sin (y-z) + sin (z-x) + 4sin((x-y)/(2)) sin( (y-z)/(2)) sin((z-x)/(2)) =0

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 x+y+z= (pi)/(2) show that sin 2x + sin 2y + sin 2z =4 cos x cosy cos z

If z_(1),z_(2)andz_(3) are the vertices of an equilasteral triangle with z_(0) as its circumcentre , then changing origin to z^(0) ,show that z_(1)^(2)+z_(2)^(2)+z_(3)^(2)=0, where z_(1),z_(2),z_(3), are new complex numbers of the vertices.

If A,B, and C are the angles of triangle,show that the system of equations x sin2A+yisnC+z sin B=0,sin C+y sin2B+z sin A=0, and x sin B+y sin A+z sin2C= posses nontrivial solution.Hence,system has infinite solutions.