# Why do these points lie in different half planes?

I'm reading Conway's complex analysis book and I'm having problems to understand one comment on page 51.

First of all, Conway begins with the definition of symmetry:

Definition: Let $\Gamma$ be a circle through points $z_2, z_3, z_4$. The points $z,z^*$ in $\mathbb C_{\infty}$ are said to be symetric with respect to $\Gamma$ if $$(z^*,z_2,z_3,z_4)=\overline{(z,z_2,z_3,z_4)}.$$

Afterwards, he begins to investigate what it means two points to be symmetric:

So my question is why do we have $z$ and $z^*$ lie in different half planes determied by $\Gamma$ if $\text{Im} \frac{z^*-z_3}{z_2-z_3}=-\text{Im}\frac{z-z_3}{z_2-z_3}$?

• What "different half planes determined by Γ" are, please? – Piquito Jan 4 '16 at 12:39

The argument of $w = \frac{z - z_3}{z_2 - z_3}$ is the oriented angle between the ray from $z_3$ to $z_2$ and the ray from $z_3$ to $z$ (the first ray being the initial side of the angle). The sign of $\operatorname{Im} w$ tells you whether this angle is between $0$ and $\pi$ or between $\pi$ and $2\pi$, hence which side of the line from $z_2$ to $z_3$ the point $z$ is on. The equation here implies that the answer to this question must be different for $z^{*}$ than it is for $z$.
• Why is the argument of $w$ the oriented angle between the ray from $z_3$ to $z_2$ and the ray from $z_3$ to $z$? thank you! – user42912 Jan 4 '16 at 12:42
• $\arg w = \arg(z - z_3) - \arg(z_2 - z_3)$. – David Jan 4 '16 at 12:44