Let `z` be not a real number such that `(1+z+z^2)//(1-z+z^2) in R ,` then prove tha `|z|=1.`

Let z_1=10+6i and z_2=4+6idot If z is any complex number such that the argument of ((z-z_1))/((z-z_2)) is pi/4, then prove that |z-7-9i|=3sqrt(2) .

Let z_(1),z_(2) and z_(3) be complex numbers such that |z_(1)|=|z_(2)|=|z_(3)|=1 then prove that |z_(1)+z_(2)+z_(3)|=|z_(1)z_(2)+z_(2)z_(3)+z_(3)z_(1)|

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_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

It z_(1) and z_(2) are two complex numbers, such that |z_(1)| = |z_(2)| , then is it necessary that z_(1) = z_(2) ?

If z_1a n dz_2 are complex numbers and u=sqrt(z_1z_2) , then prove that |z_1|+|z_2|=|(z_1+z_2)/2+u|+|(z_1+z_2)/2-u|

If z_1a n dz_2 are two complex numbers such that |z_1|=|z_2|a n d arg(z_1)+a r g(z_2)=pi , then show that z_1,=-( barz )_2dot

For any two complex number z_(1) and z_(2) , such that |z_(1)| = |z_(2)| = 1 and z_(1) z_(2) ne -1 , then show that (z_(1) + z_(2))/(1 + z_(1)z_(2)) is real number.