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单词 NsectionOfLineSegmentWithCompassAndStraightedge
释义

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

A1B1A2B2An-1Bn-1AnB

(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.

pABB1B2A1A2A3

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

AB2=23AB;B2B=13AB.

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

AB1=13AB;B1B2=13AB.

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

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更新时间:2025/5/4 3:57:24