Pasch’s theorem

Theorem.

(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 $\ell$ . ∎

Remark A companion property states that if line $\ell$ passes through one vertex $a$ of a triangle $\triangle abc$ and at least one other point on $\triangle abc$ , then it must intersect exactly one of the following:

b\mbox{, }\qquad\qquad c\mbox{, }\qquad\qquad\overline{bc}.

Of course, if $\ell$ passes through $b$ , $\overline{ab}$ must lie on $\ell$ . Similarly, $\overline{ac}$ lies on $\ell$ if $\ell$ passes through $c$ .

Title	Pasch’s theorem
Canonical name	PaschsTheorem
Date of creation	2013-03-22 15:32:09
Last modified on	2013-03-22 15:32:09
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	13
Author	CWoo (3771)
Entry type	Theorem
Classification	msc 51G05
Related topic	Angle
Related topic	OrderedGeometry