Demonstration of the impossibility to draw a parallel through a point using only a straightedge.

Question

From the responses to this question , it appears to be well know that it is impossible to trace a parallel to a straight line: $\ell$ through a point: $P$, using exclusively a straightedge.

Can you provide a demonstration of such fact?

Answer 1

A construction that uses only a straightedge can be transformed via a projective transformation (aka homography).

Suppose that you had a straightedge construction for a line $m$ through point $P$ parallel to line $\ell$. Suppose a projective transformation maps $P\rightarrow P'$ and $\ell\rightarrow \ell'$. Then the same construction would produce a line $m'$ which in general is not parallel to $\ell'$. So we have a contradiction, and there is no such straightedge construction.

The demonstration is a little more compelling if the projective transformation leaves $P$ and $\ell$ invariant. In that case, the same construction would produce two different lines, when applied before and after to the same point and line.