Pasch's theorem

Pasch's theorem states that a line entering a triangle through one side, without passing through a vertex, must exit through another side. It is a basic incidence axiom in some foundations of geometry.

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Moritz Pasch: A German mathematician whose 1882 work helped establish a rigorous, axiomatic approach to geometry. His treatment of the theorem helped make explicit the assumptions hidden in Euclidean diagrams.
Incidence geometry: The study of geometric structures defined by incidence relations between points, lines, and other objects. Pasch's theorem supplements incidence with a constraint on how a line traverses a triangle.
Order geometry: A branch of geometry studying relations such as betweenness and separation without relying on distance or angle. Pasch's theorem is a foundational rule governing separation by lines.
Euclidean geometry: Geometry based on Euclid's parallel postulate and the familiar properties of points, lines, and planes. Pasch's theorem is an order principle used in rigorous formulations of Euclidean geometry.
Axiomatic method: A method of organizing mathematics by deriving results from stated axioms and rules of inference. Pasch's theorem illustrates how geometry depends on explicit assumptions beyond incidence and order.
Betweenness: A relation expressing that one point lies between two other points on a line. The theorem's usual formal statements use betweenness to specify entry and exit.
Plane separation axiom: An axiom stating that a line divides the points outside it into two separated half-planes. This principle captures the separation behavior that underlies triangle-crossing arguments.
Projective geometry: Geometry studying incidence and cross-ratios, often by adjoining points at infinity to parallel lines. Its treatment of lines and boundaries differs from the ordered, triangle-crossing setting of Pasch's theorem.
Foundations of geometry: The study of the axioms and logical structures underlying geometric systems. The theorem is one of the principles used to formalize geometric order.
Triangle: A polygon formed by three noncollinear points joined by three line segments. Its three sides define the boundary that the entering line must cross again.