# Why is $w$ real if z is on the circle through $z_1,z_2,z_3$?

Question on answer to this (*), a duplicate of this and related to this

What I understand:

1. Based on Exer 2.14 or Exer 3.10, $$T(z)$$ refers to an arbitrary Möbius of the form $$\frac{az+b}{cz+d}, ad-bc \ne 0.$$

2. $$\forall T$$, Möbius, $$[z,z_1,z_2,z_3] = [T(z),T(z_1),T(z_2),T(z_3)]$$

3. Recall that the cross ratio itself is Möbius by Prop 3.12. Choose $$T(z)$$ to be $$[z,z_1,z_2,z_3]$$. Observe that $$T(z_1)=0, T(z_2)=1, T(z_3)=\infty,$$ which is pointed out in Prop 3.12.

4. Denote $$w:=T(z)$$. Observe that $$w = T(z) = [z,z_1,z_2,z_3] = [T(z),T(z_1),T(z_2),T(z_3)] = [w,0,1,\infty].$$

5. By (4), $$w \in \mathbb R \iff [z,z_1,z_2,z_3] \in \mathbb R$$

6. Recall that $$\forall T$$, Möbius, $$T$$ is bijective and maps clines to clines by Prop 3.1 and Thm 3.4.

7. By (6), $$w \in \mathbb R \iff z \in C(z_1,z_2,z_3)$$

8. By (5) and (7), $$\therefore, [z,z_1,z_2,z_3] \in \mathbb R \iff z \in C(z_1,z_2,z_3)$$ QED

Questions:

1. Which part of the paraphrasing is wrong, and why?

2. Why is (7) true please?

I believe this is the crux of my misunderstanding. I don't believe I understand the explanations in the answers in the linked questions.

1. Actually, it seems like we needed to have proved that $$[z,z_1,z_2,z_3] \in \mathbb R \cup \{\infty\} \iff z \in C(z_1,z_2,z_3).$$ Am I wrong? Was that done? If $$[z,z_1,z_2,z_3] = \infty$$, then $$z=z_3 \in C(z_1,z_2,z_3)$$. Not sure about converse. Oh actually, I think what may have been meant by 'real line' is $$\mathbb R \cup \{\infty\}$$? In which case, I think #4 in the answer should have been 'iff w is on the real line' rather than 'w is real' ?

2. Btw, can we get somewhere with using Cauchy-Riemann? I'm thinking if $$f$$ is real-valued then it is constant wherever it is holomorphic by Exer 2.19. Then the cross ratio $$[z,z_1,z_2,z_3]$$ equals some constant real number $$c$$. If we would solve for $$z$$ in terms of $$c,z_1,z_2,z_3$$, then could we do something with it? I mean, other than manually computing the equation of the circle and then plugging $$z$$ into it.

• Functions of the form that $T$ has send the set of circles and lines to the set of circles and lines. Since $z$ is in the circle passing through $z_1,z_2,z_3$, and these three points are sent to the real line, to the points $0,1,\infty$, then the point $z$ would also be sent to that line.
– user578878
Jul 28, 2018 at 13:49
• @nextpuzzle thanks....soooo if the 3 points' image is in R then the circle between them has an image in R and so if z is on the circle, then the image of z is real valued? Why/why not? And what about the converse? If w is in R, then why is the preimage in the circle passing through the preimages of 0,1,infty?
– BCLC
Jul 28, 2018 at 17:14
• @nextpuzzle I think I got it. Is my answer wrong please?
– BCLC
Jul 29, 2018 at 5:37

$\leftarrow$
If z is on the circle, then the image under T is on the real line. Why? T maps 3 distinct points to 2 finite points and $\infty$. Thus, T maps the points to a subset of the real line. I conclude this is the whole real line either because $\infty$ is there or because an empty or proper subset would not be a line. Now because T maps clines to clines bijectively, it maps the entire circle passing through the 3 points to entire real line. $\therefore$, if z is in the circle, its image is on the real line. But the image under T is the cross ratio.
$\rightarrow$