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.
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.
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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.