In Euclidean geometry, every set with $2$ points is collinear. My question is, are singleton sets and the empty set also collinear? I think it depends on the exact definition of collinear.
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2(i) As you say, it depends on the definition (like all of mathematics! :). (ii) In my experience, the term collinear means lying on some line, but in practice is never used except for sets containing more than one point, in which case a functionally equivalent definition would be lying on a unique line. If you ask me, yes singletons are technically collinear. (iii) In the hope it sheds light, coplanar similarly means lying in some plane. Thus, collinear points are coplanar; I don't recall ever seeing coplanar used to mean "there is a unique plane containing the points." – Andrew D. Hwang Oct 31 '23 at 12:33
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1@AndrewD.Hwang I don't think collinearity implies a unique common line, either. Suppose some construction produces three collinear points, similarly to the intersections of opposite sides of a hexagon inscribed in a conic (Pascal's theorem). Suppose then you can choose an initial setup for the construction so that the resulting three points coincide. Would we say they are no longer collinear? OK, strictly speaking, we get only one point then, just three times. But would anybody make an exception like 'the three resulting points are collinear except when they coincide'...? – CiaPan Oct 31 '23 at 13:27