# How do $w_2,w_3,w_4$ determine a circle $\Gamma'$? In page no 49 of John B Conway's "Function of One Complex Variable" I fail to understand the theorem 3.14. In particular would anybody please explain me why do $w_2,w_3,w_4$ determine a circle $\Gamma'$? I have undestood that they are distinct but why are they describing a circle? It may be any curve in $\mathbb C_{\infty}$.

I am struggling hard to understand it but couldn't manage to find any reason. Please help me.

Thank you in advance.

• "$w_2$, $w_3$, $w_4$ determine a circle" means that there is a unique circle which passes through $w_2$, $w_3$, $w_4$. Note also that $S$ is injective and therefore $w_2$, $w_3$, $w_4$ are distinct. – Robert Z Nov 13 '17 at 7:56
• But there may not be any circle passing through $w_2,w_3,w_4$. It may so happen that they are lying on a parabola. Why is it not the case? – Arnab Chatterjee. Nov 13 '17 at 7:58
• Three distinct points give you always a circle (or a line). – Robert Z Nov 13 '17 at 8:00
• If I take three distinct points from a parabola then what will happen? – Arnab Chatterjee. Nov 13 '17 at 8:01
• A parabola and a circle can have three points in common... See the last property: en.wikipedia.org/wiki/Circle#Properties – Robert Z Nov 13 '17 at 8:03