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`

In A B C ,A(z_1),B(z_2),a n dC(z_3) are inscribed in the circle |z|=5. If H(z_n) be the orthocenrter of triangle A B C , then find z_ndot

If z_1, z_2, z_3 are three complex numbers such that 5z_1-13 z_2+8z_3=0, then prove that [(z_1,(bar z )_1, 1),(z_2,(bar z )_2 ,1),(z_3,(bar z )_3 ,1)]=0

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

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

If z_1 and z_2 are two complex number such that |z_1|<1<|z_2| then prove that |(1-z_1 bar z_2)/(z_1-z_2)|<1

For any complex numbers z_1,z_2 and z_3, z_3 Im(bar(z_2)z_3) +z_2Im(bar(z_3)z_1) + z_1 Im(bar(z_1)z_2) is

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:

A(z_1),B(z_2),C(z_3) are the vertices of the triangle A B C (in clockwise). If /_A B C=pi//4 and A B=sqrt(2)(B C) , then prove that z_2=z_3+i(z_1-z_3)dot

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