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Prove: If $[z_1,z_2,z_3,z_4] \in \mathbb{R} \cup \{\infty \} $, then $z_1,z_2,z_3,z_4$ are either concyclic or collinear.

My proof below uses the geometric interpretation of cross ratio. I am not sure if it is rigorous. I am also aware of another proof which uses the fact that any circles or straight lines will be mapped to circles or straight lines under Möbius transformation. But I want to prove the clines mapping to clines theorem from cross ratio, so I cannot use this as a fact.

Cross ratio is defined as $$[z_1,z_2,z_3,z_4]= \frac{z_1 - z_3}{z_2-z_3} \frac{z_2-z_4}{z_1-z_4}$$ It is the complex number obtained from $\frac{z_1-z_3}{z_2-z_3}$ divided by $\frac{z_1-z_4}{z_2-z_4}$.

We can write $ \frac{z_1-z_3}{z_2-z_3}= r_1 e^{i \theta_1}$ where $r_1$ is the modulus of the complex ratio and $|\theta_1|= \angle z_1z_3z_2$, and similar for the other ratio.

So we can write $ [z_1,z_2,z_3,z_4]= \frac{r_1}{r_2} e^{i(\theta_1-\theta_2)}$

The value of cross ratio can only be purely real if $e^{i( \theta_1-\theta_2)}=\pm 1$ since $\frac{r_1}{r_2}$ must be real, which means the principal argument of $ \theta_1-\theta_2$ equals $0$ or $\pi$. (take principal arguments $\in (-\pi,\pi]$)

Now since the cross ratio depends on the orientations of the complex numbers (how they are arranged). We need to look into different cases.

For now consider the cases that any $3$ of the points are not collinear. Since the line passing through $z_1,z_2$ separates the complex plane into two sides, we consider two cases:

First case: $z_3,z_4$ both lie on the same side. Geometrically, $|\theta_1|,|\theta_2|$ are angles in the same segment, as they have the same sign. This corresponds to the case where $\theta_1=\theta_2$.

Second case: $z_3, z_4$ lie on different side. Then $\theta_1$ and $\theta_2$ are of different signs. So $\theta_1-\theta_2=\pi$ implies that $|\theta_1|+|\theta_2|=\pi$. Which corresponds to the opposite angles of a cyclic quadrilateral.

Both cases, the four points are concyclic. So now we consider the case where three of them lie on a straight line. To prove this, we fix any $3$ complex numbers in the formula of cross ratio and observe that (for example fix $z_1,z_2,z_3$), $\frac{z_4-z_2}{z_4-z_1}= k$ for some real $k$. So this means $z_1,z_2,z_4$ are collinear. This completes the proof.

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