n-section of line segment with compass and straightedge

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

. Draw a half-line p beginning from A but not parallelMathworldPlanetmathPlanetmath to AB. From p separate n consecutive equally long segments AA1, A1A2, A2A3, …, An-1An. Draw the line AnB and denote by B1, B2, …, Bn-1 the points of AB such that


(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 A1B1, A2B2 and A3B, and thus the corresponding angles (http://planetmath.org/CorrespondingAnglesInTransversalCutting) A1B1A, A2B2A and A3BA are equal. Similarly the angles AA1B1, AA2B2 and AA3B are equal. Because of the equal angles, the triangleMathworldPlanetmath AA2B2 is similarMathworldPlanetmathPlanetmath to the triangle AA3B with the ratio of similarity 2:3. Therefore


Also the triangle AA1B1 is similar to the triangle AA3B with the line ratio 1:3, whence


The equations show that the points B1 and B2 divide the line segment AB in 3 equal segments.

Title n-sectionMathworldPlanetmath 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