Axiom:Pasch's Axiom (Euclidean Geometry)

This page is about Pasch's Axiom in Euclidean Geometry. For other uses, see Axiom: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 line intersects two of the triangle's edges 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.

He published this axiom in 1882, thereby showing that Euclid's postulates are incomplete.

Also see

• Pasch's Axiom in Tarski's Geometry