# Pasch's theorem

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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.

## Statement[edit]

The statement is as follows:

**Pasch's theorem** — 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.]

## Notes[edit]

**^**Pasch 1912**^**Coxeter (1969, p. 179) states the result in 12.274 but does not refer to it specifically as Pasch's theorem.

## References[edit]

- Coxeter, H.S.M. (1969),
*Introduction to geometry*(2nd ed.), John Wiley and Sons, ISBN 978-0-471-18283-2, Zbl 0181.48101 - Pasch, Moritz (1912) [first edition 1882],
*Vorlesungen uber neuere Geometrie*(in German) (2nd ed.), Leipzig: B.G. Teubner