The following is a theorem and a proof from Complex Variables by Herb Silverman. The bolded are parts that I don't understand in the proof.
Theorem: Given three distinct points, $z_1,z_2, z_3$ in the extended $z$ plane and three distinct points $w_1,w_2,w_3$ in the extended $w$ plane, there exists a unique bilinear transformation $w=t(z)$ such that $T(z_k)=w_k$ for $k=1,2,3$.
Proof. We first assume that none of the six points is $\infty$. Let $w=T(z)=\frac{az+b}{cz+d}$. We wish to solve for $a,b,c,$ and $d$ in terms of $z_1,z_2,z_3,w_1,w_2,$ and $w_3$. This sounds more complicated than it is. For $k=1,2,3,$ we have
$$w-w_k=\frac{az+b}{cz+d}-\frac{az_k+b}{cz_k+d}=\frac{(ad-bc)(z-z_k)}{(cz+d)(cz_k+d)}(3.9)$$
From (3.9) we obtain
$$\frac{w-w_1}{w-w_3}=\frac{cz_1+d}{cz_3+d}\frac{z_2-z_3}{z_2-z_1} (3.10)$$
Replacing $z$ by $z_2$ and $w$ by $w_2$ in (3.10) leads to
$$\frac{w_2-w_3}{w_2-w_1}=\frac{cz_1+d}{cz_3+d}\frac{z_2-z_3}{z_2-z_1} (3.11)$$
Multiplying (3.10) by (3.11) we have
$$\frac{(w-w_1)(w_2-w_3)}{(w-w_3)(w_2-w_1)}=\frac{(z-z_1)(z_2-z_3)}{(z-z_3)(z_2-z_1)} (3.12)$$
Solving for $w$ in terms of $z$ and the six points gives the desired transformation. If one of the points were the point at $\infty$, say $z_3=\infty$, (3.12) would be modified by taking the limit as $z_3$ approached $\infty$. In this case, we would have
$$\frac{(w-w_1)(w_2-w_3)}{(w-w_3)(w_2-w_1)}=\frac{z-z_1}{z_2-z_1} (3.13)$$
Corollary. Given three distinct points, $z_1,z_2,z_3$ in the extended $z$ plane there exists a unique bilinear transformation $w=T(z)$ such that $T(z_1)=0, T(z_2)=1, T(z_3)=\infty$ and it is given by
$$w=\frac{(z-z_1)(z_2-z_3)}{(z-z_3)(z_2-z_1)}$$.
Firstly, I don't see how to obtain the desired transformation. I did find an expression for $w$ in terms of all other variables (which looks complicated) and found $T(z_1)=w_1$ and $T(z_2)=w_2$. However, I don't get $T(z_3)=w_3$. In fact, if we look at (3.12) and plug in $z_3$ in place of $z$, the fraction on the right hand side isn't even defined since the denominator is $0$…So I don't see how we can get the desired result from this expression. Actually, I don't understand why the author sets out to find an expression for $w$ by multiplying fractions. What may be the reasoning behind this?
Moreover, I don't understand the part when one of the points is $\infty$. It makes sense to take the limit as $z_3 \to \infty$, but how does this bring the expression (3.13)?
Finally, for the corollary, again this is similar to the first question, I think we're supposed to use (3.12) and simply plug in the appropriate $w_k$ values but in case of $T(z_3)=\infty$, the expression just doesn't make sense to me. How does $\frac{(w-w_1)(w_2-\infty)}{(w-\infty)(w_2-w_1)}$ make sense?
I would greatly appreciate it if anyone clarifies the above questions to me, I'm having trouble reading this page because of these.
Best Answer
I have not read all your post, but perhaps a nicer way of proving the result is to start with the corollary: first prove that there is a unique linear fractional transformations which maps 3 distinct points to (0,1,$\infty$). It is not difficult to write down the expression (which you have), and for unicity you can use e.g. the fact that any linear fractional transformation which fixes more than 2 points is the identity.
Now recall that all linear fractional transformations are invertible, therefore you also know that you can map (0,1,$\infty$) to any triple of distinct points.
Putting the two things together you can easily prove that it is possible to map any triple of distinct points to any other triple of distinct points.