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[parent] $n$-section of line segment with compass and straightedge (Algorithm)

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

Solution. Draw a half-line $ p$ beginning from $ A$ but not parallel to $ AB$. From $ p$ separate $ n$ consecutive equally long segments $ AA_1$, $ A_1A_2$, $ A_2A_3$, ..., $ A_{n-1}A_n$. Draw the line $ A_nB$ and denote by $ B_1$, $ B_2$, ..., $ B_{n-1}$ the points of $ AB$ such that

$\displaystyle A_1B_1\;\parallel\;A_2B_2\;\parallel\;\ldots\;\parallel\;A_{n-1}B_{n-1}\;\parallel\;A_nB$
(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$.


\begin{pspicture}(-0.5,-0.5)(6.5,4) \psline[linecolor=blue](0,0)(6,0) \psline(0,... ...,1.05){$A_1$} \rput[l](2.1,1.85){$A_2$} \rput[l](3.4,2.7){$A_3$} \end{pspicture}
The line $ AB$ intersects the parallel lines $ A_1B_1$, $ A_2B_2$ and $ A_3B$, and thus the corresponding angles $ A_1B_1A$, $ A_2B_2A$ and $ A_3BA$ are equal. Similarly the angles $ AA_1B_1$, $ AA_2B_2$ and $ AA_3B$ are equal. Because of the equal angles, the triangle $ AA_2B_2$ is similar to the triangle $ AA_3B$ with the ratio of similarity $ 2\!:\!3$. Therefore
$\displaystyle AB_2 = \frac{2}{3}AB;\quad B_2B = \frac{1}{3}AB.$
Also the triangle $ AA_1B_1$ is similar to the triangle $ AA_3B$ with the line ratio $ 1\!:\!3$, whence
$\displaystyle AB_1 = \frac{1}{3}AB;\quad B_1B_2 = \frac{1}{3}AB.$
The equations show that the points $ B_1$ and $ B_2$ divide the line segment $ AB$ in 3 equal segments.



"$n$-section of line segment with compass and straightedge" is owned by pahio.
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Cross-references: equations, line ratio, ratio of similarity, similar, triangle, angles, parallel lines, intersects, compass and straightedge construction of parallel line, points, line, segments, consecutive, parallel, divide, integer, positive, line segment
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This is version 8 of $n$-section of line segment with compass and straightedge, born on 2007-07-21, modified 2007-08-15.
Object id is 9784, canonical name is NSectionOfLineSegmentWithCompassAndStraightedge.
Accessed 687 times total.

Classification:
AMS MSC: 51-00 (Geometry :: General reference works )
 51M05 (Geometry :: Real and complex geometry :: Euclidean geometries and generalizations)
 51F99 (Geometry :: Metric geometry :: Miscellaneous)
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