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Given $z_1,z_2,z_3,z_4$ 4 different points of $ {\mathbb C}$, we define the cross ratio $(z_1,z_2,z_3,z_4)$ as

$(z_1,z_2,z_3,z_4)\longrightarrow [z_1,z_2,z_3,z_4]$

As a first step, we had to show that there is an invariance of cross ratio under Möbius transformation , what I did.

I'm stuck here , how can I prove :

$$[z_1,z_2,z_3,z_4]=\frac{(\frac{z_1-z_3}{z_2-z_3})}{(\frac{z_1-z_4}{z_2-z_4})}$$

And I found this following statement

$ABC$ a triangle in complex plane & $\;P,Q,R\;$ 3 points such that :

$P \in (BC) \; , \;Q \in (CA)\;,\; R \in (AB) $

$(QR)\; $and $(BC)$ are concurrent at a point D

We have :

$$\frac{\overline{PC}}{\overline{PB}} \, .\frac{\overline{QA}}{\overline{QC}} \, . \frac{\overline{RB}}{\overline{RA}}=[C,B,A',D]$$

How can I prove it ?

Thanks in advance for your help.

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    $\begingroup$ brb time to dig through my hw $\endgroup$ – Saketh Malyala Dec 3 at 1:19
  • $\begingroup$ You haven't given a definition of cross-ratio!!! You just wrote square brackets. $\endgroup$ – Ted Shifrin Dec 3 at 2:07
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I'm not sure if this serves as an answer, but I wanted to post these images, which I pulled straight out of my homework.

enter image description here

enter image description here

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    $\begingroup$ fundamentals of complex analysis by eb saff&snider, ch7-4 exercise #4 $\endgroup$ – Saketh Malyala Dec 3 at 1:23
  • $\begingroup$ It helps thanks a lot , no idea for the 2nd part ? $\endgroup$ – SkyLand77 Dec 3 at 1:29

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