Let $p_1,...,p_{n+2}$ and $q_1,...,q_{n+2}$ be two sets of distinct points in general position in $\Bbb P^n$ (there may however be overlaps between the two sets.). Then there exists $\phi\in Aut(\mathbb{P}^n)$ taking $p_i\mapsto q_i$.
I've been having trouble with what should be the extremely simple proof of this fact. Michael Artin proves it for $n=2$ (though the same idea should work for any $n$) here http://math.mit.edu/classes/18.721/chpcurvev5.pdf following the line of reasoning I was trying to use. My confusion arises when he says to adjust the $p_i$ by a factor of $1/c_1$. This sounds like a perfectly fine thing to do in $\Bbb P^n$, but I don't see how it leads to the conclusion that $q=p_0+p_1+p_2$. If there is a typo, and he meant to write that we adjust each $p_i$ by $1/c_i$, then I understand how he reaches the conclusion, but I don't see how we are allowed to separately scale three tuples of coordinates before adding them together, i.e. Artin seems to be jumping back-and-forth between projective and affine coordinates without justification. Do I misunderstand the nature of $PGL(n)$ in some way?