Pasch’s theorem


(Pasch) Let abc be a triangleMathworldPlanetmath with non-collinear vertices a,b,c in a linear ordered geometry. Suppose a line intersects one side, say open line segmentPlanetmathPlanetmath ab¯, at a point strictly between a and b, then also intersects exactly one of the following:

bc¯    ac¯    c.

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

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

b    c    bc¯.

Of course, if passes through b, ab¯ must lie on . Similarly, ac¯ lies on if 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