Pasch's theorem 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, is a result in plane geometry which cannot be derived from Euclid's postulates.
Contents 1 Statement 2 See also 3 Notes 4 References 5 External links 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). [Here, for example, (a, b, c) means that point b lies between points a and c.] See also Ordered geometry Pasch's axiom Notes ^ Pasch 1912 ^ 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 External links Weisstein, Eric W. "Pasch's Theorem". MathWorld. This elementary geometry-related article is a stub. You can help Wikipedia by expanding it.
Categories: Euclidean plane geometryFoundations of geometryOrder theoryTheorems in plane geometryElementary geometry stubs
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