I am reading Schwartz's book "A mathematician grappling with his century". On page 54 he writes
I made an astute discovery of the following theorem: There is a point in the plane through which every real line in the plane passes. The theorem is obviously wrong, but it is quite extraordinary.
He then goes on to give a "proof" which of the above theorem. I am not familiar with the terminology of the proof. So I quote the "proof" below, can someone explain to me the argument?
Let $D, D'$ be two complex conjugate lines. Their intersection point $O$ is real. If $M, M'$ are distinct complex conjugate numbers on $D, D'$ respectively, then the line $M, M'$ is real. Conversely, every real line cuts $D$ and $D'$ in mutually complex conjugate points. Thus we have an algebraic bijection between the points of two lines $D$ and $D'.$ So by a well-known theorem, it is homographic. But, if there is a homographic correspondence $(M, M')$ between the points of two lines, and if the point of intersection of two lines $O$ correspond to itself, which is the case since it is real, then the line $MM'$ passes through a fixed point. So, all real lines passed through some fixed point.
Now, my questions are as follows?
a) By complex conjugate lines does he mean $D=a+tb$ and $D'=\bar{a}+t\bar{b}$ for some complex numbers $a, b?$
b) What does he mean by $MM'$ is real? Does it mean to parallel to $y$-axis?
c) What is an algebraic bijection? Is it just a one-to-one correspondence?
d) Which well-known theorem is he referring to? And what is a homographic correspondence?
Overall, can someone run through the argument being made here (even though it is obviously false) in a contemporary language which I can understand?