Pasch's theorem

In geometry, Pasch's theorem, stated in 1882, is a result of plane geometry which cannot be derived from Euclid's postulates. It would now be considered as order theory, but the point it makes is in relation to the axiomatic method.

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).

Moritz Pasch


* MathWorld page

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