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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 $\text{Im} \frac{z^*-z_3}{z_2-z_3}=-\text{Im}\frac{z-z_3}{z_2-z_3}$?

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  • $\begingroup$ What "different half planes determined by Γ" are, please? $\endgroup$ – Piquito Jan 4 '16 at 12:39
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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 $z$.

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  • $\begingroup$ 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! $\endgroup$ – user42912 Jan 4 '16 at 12:42
  • $\begingroup$ $\arg w = \arg(z - z_3) - \arg(z_2 - z_3)$. $\endgroup$ – David Jan 4 '16 at 12:44

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