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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?

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  • $\begingroup$ If true, I imagine it is difficult to prove. $\endgroup$ Nov 29, 2020 at 14:58
  • $\begingroup$ 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. $\endgroup$
    – David K
    Nov 29, 2020 at 15:05
  • $\begingroup$ I said "using exclusively a straightedge". $\endgroup$ Nov 29, 2020 at 15:08
  • $\begingroup$ math.stackexchange.com/q/96170/139123 may be of interest. $\endgroup$
    – David K
    Nov 29, 2020 at 15:31
  • $\begingroup$ Yes it is very interesting, thank you! $\endgroup$ Nov 29, 2020 at 21:59

1 Answer 1

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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.

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  • $\begingroup$ I felt almost satisfied. Anyway. I have two questions: i) I wonder if decisions between steps, like for example: to put an arbitrary point on one side or on the other of a line depending of the previous state of the construction are also invariant under projective maps. ii) what if infinite steps are allowed? In this case it appears that some construction is possible. $\endgroup$ Nov 29, 2020 at 22:08
  • $\begingroup$ Good question. i) I think the only thing that is relevant in the state of construction is the arrangement of points and lines and their incidences. Things like angle and distance don't come into play, because you have only a straightedge, not a ruler or protractor. So points/lines/incidences/sidedness would be projectively invariant. ii) I'm not sure what infinite steps buys you, or whether it should even be considered. You'd have to elaborate. $\endgroup$
    – brainjam
    Nov 29, 2020 at 22:35
  • $\begingroup$ An obvious way to do in infinite steps is as follow. 1) Choose a point Q1 on ℓ and trace the line Q1P. 2) Choose a point Q2 at the right of Q1 on ℓ and trace the line Q2P. 3) etc ... The infinite N line QNP is parallel to ℓ. This method requires a infinite amount of paper, but I wonder if perhaps is possible to restrict the construction to a finite region using projective methods like this $\endgroup$ Nov 30, 2020 at 6:28
  • $\begingroup$ @Perspectiva8, Yes, and you can get infinitely close to the center of the circle this way (quora.com/…). But in either infinite construction, you never actually get there. And by this line of thinking you could also trisect an angle with ruler and compass (generally held to be impossible). So the short answer is infinite constructions don't work, but if you want to pursue it please open another question dealing specifically with infinite constructions. $\endgroup$
    – brainjam
    Nov 30, 2020 at 15:15
  • $\begingroup$ Basically I agree with your comment in the meaning , but not in the words. Infinite constructions, in fact do work. Perhaps you should say "infinite constructions are not allowed" $\endgroup$ Jan 7, 2021 at 12:24

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