Pasch axiom

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One of the order axioms in the Hilbert system of axioms of Euclidean geometry. The statement of the axiom uses the concept "lies within (between) a segment" , and a segment is regarded here as a system of two distinct points $A$ and $B$; points lying "between" $A$ and $B$ are said to be points (or interior points) of the segment. The concept "between" (lying between) is described by the group of axioms of order, to which Pasch' axiom also belongs. It can be stated as follows: Let $A$, $B$ and $C$ be three non-collinear points and let $a$ be a straight line in the plane $(ABC)$ of these three points that does not pass through any of them; if the line passes through a point between $A$ and $B$, then it must also pass through a point between $A$ and $C$ or a point between $B$ and $C$.

This is an axiom of absolute geometry. By means of Hilbert's other axioms of order it can be proved that the line $a$ cannot intersect both $AC$ and $BC$. M. Pasch stated this axiom in [1].


[1] M. Pasch, "Vorlesungen über neuere Geometrie" , Springer, reprint (1926)
[2] D. Hilbert, "Grundlagen der Geometrie" , Teubner, reprint (1962)


Sometimes the Veblen–Young axiom of projective geometry is, erroneously, called Pasch' axiom.


[a1] H.S.M. Coxeter, "Introduction to geometry" , Wiley (1969) pp. 178
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This article was adapted from an original article by L.A. Sidorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article