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  "body": "\n## Statement\n\nLet $A$, $B$ and $C$ be three non-collinear points of a plane.\nLet $\\ell$ be a [[line|0MP9BCF6DP]] of that plane that passes through none of them.\nSuppose that $\\ell$ contains a point [[between|CE8GX6ADZS]] $A$ and $B$.\nThen $\\ell$ contains a point between $A$ and $C$ or a point between $B$ and $C$.\n\n## Remarks\n\nMoritz Pasch stated it as an axiom in 1882, after he noticed that Euclid uses it without saying so.\nIn Hilbert's system it is one of the axioms of order, and it cannot be proved from the other axioms of incidence and order.\nFrom the axioms of order one can also prove that $\\ell$ cannot meet all three sides of the [[triangle|0CTNFGYMM2]] $ABC$.\nIn $\\R^2$ it holds: write $\\ell = \\set{x : f(x) = 0}$ with $f$ affine, so that $f(A)$ and $f(B)$ have opposite signs.\nThen $f(C)$ has the sign opposite to one of them, and $f$ vanishes between $C$ and that point.\nIt is the basis of the plane separation theorem: a line divides the rest of the plane into two half-planes.\n\n## Explanations\n\n### L1 human\n\nIf a straight line goes into a triangle through one side, it has to come out through another side.\nA picture makes this obvious, but it does not follow from Euclid's postulates.\nSo modern axiom systems state it as an axiom.\n\n### L2 human\n\nPasch'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.\nMany proofs in Euclid use it without saying so.\nThe crossbar theorem, which says that a ray inside an angle of a triangle meets the opposite side, rests on it.\n",
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