Pasch's axiom

Remarks

Moritz Pasch stated it as an axiom in 1882, after he noticed that Euclid uses it without saying so. In Hilbert's system it is one of the axioms of order, and it cannot be proved from the other axioms of incidence and order. From the axioms of order one can also prove that $\ell$ cannot meet all three sides of the triangle $ABC$. In $\R^2$ it holds: write $\ell = \set{x : f(x) = 0}$ with $f$ affine, so that $f(A)$ and $f(B)$ have opposite signs. Then $f(C)$ has the sign opposite to one of them, and $f$ vanishes between $C$ and that point. It is the basis of the plane separation theorem: a line divides the rest of the plane into two half-planes.

Going deeper

For a student

Pasch's axiom is what one uses to prove that a line splits the plane into two sides, and that the inside of a triangle or an angle is well defined. Many proofs in Euclid use it without saying so. The crossbar theorem, which says that a ray inside an angle of a triangle meets the opposite side, rests on it.