Axiom:Pasch's Axiom (Euclidean Geometry)

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 line intersects two of the triangle's edges or none.

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

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

Also see

 * Pasch's Axiom in Tarski's Geometry