# Pasch’s theorem

###### Theorem.

(Pasch) Let $\mathrm{\u25b3}\mathit{}a\mathit{}b\mathit{}c$ be a triangle^{} with
non-collinear vertices $a\mathrm{,}b\mathrm{,}c$ in a linear ordered geometry.
Suppose a line $\mathrm{\ell}$ intersects one side, say open line segment^{} $\overline{a\mathit{}b}$, at a
point strictly between $a$ and $b$, then $\mathrm{\ell}$ also intersects exactly one of the following:

$$\overline{bc}\mathit{\text{,}}\mathit{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}\overline{ac}\mathit{\text{,}}\mathit{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}c.$$ |

###### Proof.

First, note that vertices $a$ and $b$ are on opposite sides of line $\mathrm{\ell}$. Then either $c$ lies on $\mathrm{\ell}$, or $c$ does not. if $c$ does not, then it must lie on either side (half plane) of $\mathrm{\ell}$. In other words, $c$ and $a$ must be on the opposite sides of $\mathrm{\ell}$, or $c$ and $b$ must be on the opposite sides of $\mathrm{\ell}$. If $c$ and $a$ are on the opposite sides, $\mathrm{\ell}$ 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 $\mathrm{\ell}$. ∎

Remark A companion property states that if line $\mathrm{\ell}$ passes through one vertex $a$ of a triangle $\mathrm{\u25b3}abc$ and at least one other point on $\mathrm{\u25b3}abc$, then it must intersect exactly one of the following:

$$b\text{,}\mathit{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}c\text{,}\mathit{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}\overline{bc}.$$ |

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