Pasch's axiom

Pasch's axiom states that a line entering a triangle through one side, without passing through a vertex, must exit through another side. It formalizes a basic interaction between betweenness and incidence in plane geometry.

Betweenness: A relation stating that one point lies between two others, often used to define order on a line. Pasch's axiom turns this order relation into a constraint on how a line meets a triangle.
Moritz Pasch: A German mathematician whose 1882 work helped establish the modern axiomatic treatment of geometry. He gave the axiom its name and made such assumptions explicit.
Convex set: A set in which the segment between any two points of the set lies entirely within it. Pasch-style order reasoning helps establish how lines interact with convex regions.
Incidence axiom: An axiom specifying basic relations between geometric objects, such as points and lines. Incidence alone does not determine the crossing behavior imposed by Pasch's axiom.
Incidence geometry: Geometry built from relations of incidence, such as which points lie on which lines. The axiom supplements incidence with an order condition about line crossings.
Vorlesungen über neuere Geometrie: Moritz Pasch's 1882 book that developed a rigorous, axiomatic approach to geometry. This work presented the foundational setting associated with Pasch's axiom.
Jordan curve theorem: The theorem that every simple closed curve in the plane separates it into an interior and an exterior. Both concern separation in the plane, though the theorem requires substantially deeper machinery.
Continuity axiom: An axiom ensuring that geometric magnitudes or points have no relevant gaps. Continuity and Pasch address distinct foundational properties of geometric planes.
Triangle: A polygon formed by three noncollinear points and the three segments joining each pair. The three sides supply the boundary whose crossings Pasch's axiom constrains.
Euclid's Elements: An ancient Greek mathematical treatise presenting geometry through definitions, postulates, and propositions. Euclid's development relies on intuitive order facts that Pasch's axiom states explicitly.

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