Axiom:Pasch's Axiom (Euclidean Geometry)

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This page is about Pasch's Axiom in the context of Euclidean Geometry. For other uses, see Pasch's Axiom.

Axiom

Let a triangle and a straight line lie in the same plane such that the line does not go through any of the vertices of the triangle.

Then if the line intersects one side of the triangle, it intersects another.

Pasch's Axiom.png

That is, such a straight line intersects two of the triangle's sides or none.


Comment

This result is used by Euclid implicitly in The Elements, but it can not be derived from the postulates.




Source of Name

This entry was named for Moritz Pasch.


Historical Note

Moritz Pasch published this axiom in $1882$, during the course of showing that Euclid's postulates are incomplete.


Also see