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

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?

• If true, I imagine it is difficult to prove. – Adam Rubinson Nov 29 '20 at 14:58
• In an answer to the question it is shown that it is possible to construct a parallel to a given straight line through a given point if you are also given some additional information to start with. – David K Nov 29 '20 at 15:05
• I said "using exclusively a straightedge". – Perspectiva8 Nov 29 '20 at 15:08
• math.stackexchange.com/q/96170/139123 may be of interest. – David K Nov 29 '20 at 15:31
• Yes it is very interesting, thank you! – Perspectiva8 Nov 29 '20 at 21:59

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.