Pasch's theorem

In geometry, Pasch's theorem, stated in 1882 by the German mathematician Moritz Pasch, is a result in plane geometry which cannot be derived from Euclid's postulates.

The statement is as follows. Given points a, b, c, and d on a line, if it is known that the points are ordered as (a, b, c) and (b, c, d), then it is also true that (a, b, d). [Here, for example, (a, b, c) means that point b lies between points a and c.]

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