If $\Gamma$ is a circle through points $z_2,z_3,z_4$ then $z^*$ and $z$ in $\Bbb C_\infty$ are said to be symmetric if
$(z^*,z_2,z_3,z_4)$= Conjugate of $(z,z_2,z_3,z_4)$ (I couldn't get an overline on this bracket)
If $\Gamma$ is a straight line then $z^*$ and $z$ should be symmetric w.r.t. $\Gamma$ if the line through $z^*$ and $z$ is perpendicular to $\Gamma$ and $z^*$ and $z$ are at the same distance from $\Gamma$ but on the opposite sides of $\Gamma$. Am I correct ?
If I am, then if both $z^*$ and $z$ lie on $\Gamma$, then that would mean that they are not symmetric ?