Pasch's theorem

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.

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

Statement

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.]

See also

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

References
  • 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
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