# Using cross ratio to prove a theorem

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 ?

• brb time to dig through my hw – Saketh Malyala Dec 3 at 1:19
• You haven't given a definition of cross-ratio!!! You just wrote square brackets. – Ted Shifrin Dec 3 at 2:07  