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This is from Kiselev's Geometry: Planimetry, page 2:

For every two points in space there is a straight line passing through them, and such a line is unique.

So I have two fixed points, and I draw a line through them. Now I make an exact copy of the line, and the copy coincides with the original. It seems like I should be allowed to say: "I have two lines passing through the same points". But the book says that a line passing through two fixed points is unique, so there seems to be a contradiction here.

If I move the copy and the original so they no longer coincide, are they still just one line, or are they two lines now?

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    $\begingroup$ If move the line such that they no longer coincide, then the line would not pass through the two points. $\endgroup$ – DHMO Apr 12 '17 at 17:38
  • $\begingroup$ Two lines which coincide are considered the same line. If you've given the first line the label $l_1$ and the second the label $l_2$, then you've essentially just labelled the same object in two different ways (which is fine but it doesn't make them distinct objects). $\endgroup$ – Dylan Apr 12 '17 at 17:39
  • $\begingroup$ @DHMO Right. So when two lines pass through the same points they are considered to be one line, but if they are moved apart they are considered to be two lines? $\endgroup$ – Ovi Apr 12 '17 at 17:40
  • $\begingroup$ @Ovi the theorem is that if you have two lines which pass through the same pair of points, then they must coincide. $\endgroup$ – DHMO Apr 12 '17 at 17:41
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    $\begingroup$ Maybe Ovi thinks that a point is an object that has a location in the plane, and can be picked up and put down at a different location, with all its other properties the same. But in Euclidean geometry is point is a location in the plane and nothing more, so it does not make sense to say you are moving it somewhere else. $\endgroup$ – MJD Apr 12 '17 at 17:55
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If you call your first line, say $g$, and your copy $g'$, then you can say that you have two lines ($g$ and $g'$) passing through the same points. But these two lines are coincide as you mentioned yourself so you do not have two distinct lines through the same points. What they mean by unique is that there is no line that does not coincide exactly with the first one and passes through the same two points.

And if you move your second line it is considered a transformation and it is now a new line distinct from the original line. That is what is meant by unique. Two lines going through the same two points are actually equal, therefore the same line. All you actually did was assign a new label or different variable, but the new variable can be shown to be equal to the original variable, therefore they are the same. There weren't two lines, to begin with, just different names for the same thing.

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