Theorem 1   (Pasch) Let $\triangle abc$ be a triangle with non-collinear vertices $a,b,c$ in a linear ordered geometry. Suppose a line $\ell$ intersects one side, say open line segment $\overline{ab}$ , at a point strictly between $a$ and $b$ , then $\ell$ also intersects exactly one of the following: $$\overline{bc}\mbox{, }\qquad\qquad\overline{ac}\mbox{, }\qquad\qquad c.$$

Proof. First, note that vertices $a$ and $b$ are on opposite sides of line $\ell$ . Then either $c$ lies on $\ell$ , or $c$ does not. if $c$ does not, then it must lie on either side (half plane) of $\ell$ . In other words, $c$ and $a$ must be on the opposite sides of $\ell$ , or $c$ and $b$ must be on the opposite sides of $\ell$ . If $c$ and $a$ are on the opposite sides, $\ell$ has a non-empty intersection with $\overline{ac}$ . But if $c$ and $a$ are on the opposite sides, then $c$ and $b$ are on the same side, which means that $\overline{bc}$ does not intersect $\ell$ .