# $n$-section of line segment with compass and straightedge

Task. Let $AB$ be a given line segment^{} and $n$ a positive integer $>1$. Divide $AB$ to $n$ equal parts.

. Draw a half-line $p$ beginning from $A$ but not parallel^{} to $AB$. From $p$ separate $n$ consecutive equally long segments $A{A}_{1}$, ${A}_{1}{A}_{2}$, ${A}_{2}{A}_{3}$, …, ${A}_{n-1}{A}_{n}$. Draw the line ${A}_{n}B$ and denote by ${B}_{1}$, ${B}_{2}$, …, ${B}_{n-1}$ the points of $AB$ such that

$${A}_{1}{B}_{1}\parallel {A}_{2}{B}_{2}\parallel \mathrm{\dots}\parallel {A}_{n-1}{B}_{n-1}\parallel {A}_{n}B$$ |

(see compass and straightedge construction of parallel line). These points divide the line segment $AB$ in $n$ equal segments.

Proof. For clarity, we prove the theorem only in the case $n=3$.

The line $AB$ intersects the parallel lines ${A}_{1}{B}_{1}$, ${A}_{2}{B}_{2}$ and ${A}_{3}B$, and thus the corresponding angles (http://planetmath.org/CorrespondingAnglesInTransversalCutting) ${A}_{1}{B}_{1}A$, ${A}_{2}{B}_{2}A$ and ${A}_{3}BA$ are equal. Similarly the angles $A{A}_{1}{B}_{1}$, $A{A}_{2}{B}_{2}$ and $A{A}_{3}B$ are equal. Because of the equal angles, the triangle^{} $A{A}_{2}{B}_{2}$ is similar^{} to the triangle $A{A}_{3}B$ with the ratio of similarity $2:3$. Therefore

$$A{B}_{2}=\frac{2}{3}AB;{B}_{2}B=\frac{1}{3}AB.$$ |

Also the triangle $A{A}_{1}{B}_{1}$ is similar to the triangle $A{A}_{3}B$ with the line ratio $1:3$, whence

$$A{B}_{1}=\frac{1}{3}AB;{B}_{1}{B}_{2}=\frac{1}{3}AB.$$ |

The equations show that the points ${B}_{1}$ and ${B}_{2}$ divide the line segment $AB$ in 3 equal segments.

Title |
$n$-section^{} of line segment with compass and straightedge |
---|---|

Canonical name | NsectionOfLineSegmentWithCompassAndStraightedge |

Date of creation | 2013-03-22 17:24:41 |

Last modified on | 2013-03-22 17:24:41 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 12 |

Author | pahio (2872) |

Entry type | Algorithm |

Classification | msc 51F99 |

Classification | msc 51M05 |

Classification | msc 51-00 |

Related topic | CompassAndStraightedgeConstructionOfParallelLine |