Pasch's theorem

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Not to be confused with Pasch's axiom regarding a line through a triangle

In geometry, Pasch's theorem, stated in 1882 by the German mathematician Moritz Pasch,[1] 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).[2] [Here, for example, (a, b, c) means that point b lies between points a and c.]

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  1. ^ Vorlesungen über neuere Geometrie (Leipzig, 1882)
  2. ^ Coxeter (1969, p. 179) states the result in 12.274 but does not refer to it specifically as Pasch's theorem.

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