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`

Let A(z_1), B(z_2) and C(z_3) forms an acute angled triangle in argand plane having origin as it’s orthocentre. Prove that z_1barz_2+bar(z)_1 z_2=z_2barz_3+barz_2z_3=z_3barz_1 +barz_3z_1

If arg (bar (z) _ (1)) = arg (z_ (2)) then

For any three complex numbers z_1, z_2 and z_3 , prove that z_1 lm (bar(z_2) z_3)+ z_2 lm (bar(z_3) z_1) + z_3lm(bar(z_1) z_2)=0 .

If z_(1) = 3+ 2i and z_(2) = 2-i then verify that (i) bar(z_(1) + z_(2)) = barz_(1) + barz_(2)

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

Let z_(1), z_(2), z_(3) be three non-zero complex numbers such that z_(1) bar(z)_(2) = z_(2) bar(z)_(3) = z_(3) bar(z)_(1) , then z_(1), z_(2), z_(3)

The conjugate of x+2bar(z) is (i) 2bar(z)+z (ii) bar(z)+2z( iii) bar(z)-2z( iv )bar(z)-z

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 .

If z_(1),z_(2),z_(3) are three complex numbers such that 5z_(1)-13z_(2)+8z_(3)=0, then prove that [[z_(1),(bar(z))_(1),1z_(2),(bar(z))_(2),1z_(3),(bar(z))_(3),1]]=0