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I know Poncelet-Steiner tells us that given a circle and its center, straightedge alone is equivalent to straightedge and compass. My question is, what can we construct with purely straightedge? We certainly can't construct any square roots in a finite number of steps. Given a segment of unit length, is it possible to construct any rational number?

Thanks in advance. I wanted to know because I wanted to show that you can construct any square root with straightedge alone in an infinite number of steps.

EDIT: What would you need to construct every rational? Would some manner of constructing parallel lines suffice? Would a segment of length 2 in addition to the unit segment suffice?

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    Given straightedge alone, what are you given to start with? 2 points? If that is the case, all you can construct is a line, since that's the only thing you can build over those two points.2012-01-03
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    With only a straightedge, you cannot make copies of your unit length segment, so it is rather useless :)2012-01-03
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    The only thing that a straightedge lets you do is connect two points to get a line (and by extension, find intersections of two lines). I think that, given a segment of length 1 and a straightedge, you can't even produce a segment of length 2, let alone the other rational numbers. (Did you mean a *ruler*, instead of a straightedge?)2012-01-03
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    (If one can construct a segment of length 2, then I think one can construct all rationals, no?)2012-01-03
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    @MarianoSuárez-Alvarez I think you also need to solve the problem of drawing parallel lines through given points....2012-01-03
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    Ah. Indeed. But it would be enough to have *one* parallel line to the one we start with.2012-01-03
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    Sorry about the badly-phrased question. This was more out of curiosity than anything else. I remember reading somewhere that, starting from scratch, you can make a projective grid, which sounded strong enough to beget all the rationals, but alas, I've forgotten how.2012-01-03

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