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Trisecting a Line Segment

© Copyright 2003, Jim Loy

I received email offering this method of trisecting a line segment. We want to trisect segment AB. We bisect it at C, draw equilateral triangle BCD, draw AD which intersects the perpendicular bisector of AB at E. We draw draw BE, extending it so BE = EF. We draw AF. We bisect angle AFB, which trisects segment AB at G. My simplified diagram is shown below that. We draw (there are ways to do this besides the way shown above) the 30-60-90 degree right triangle shown, and bisect the 60 degree angle. AB and FG are not only bisectors of two angles of an equilateral triangle, but are medians (see The Centers of a Triangle) of the same equilateral triangle, and medians trisect each other.

Trisecting a line segment, using compasses and straightedge (see Geometric Constructions) is easy. Here is the standard method. We want to trisect AB. We draw an arbitrary line segment through A, and duplicate it twice, producing a trisected line segment AC. We draw BC. Then through the other two points on AC (the trisecting points), we draw lines parallel to BC. These lines trisect AB. This method can be used to divide a segment into any number of equal segments.

That method can also be viewed as a way to trisect using similar triangles. There are other ways to use similar triangles, one three times larger than another, to trisect a side of the larger triangle.

Speaking of medians, we don't have to use equilateral triangles. Here I want to trisect AB. I draw an arbitrary segment AC, and extend it to D, so that AD=AC. I bisect BD at E. CE trisects AB at F.

Here is a related method. We want to trisect AB. We draw the perpendicular bisector CD, choosing an arbitrary point D on it. We bisect AD at E, bisect BE at F, and DF trisects AB.

Addendum:

A reader named TR sent me this method, which uses similar triangles (the diagonal lines through A and B are parallel, and the points on these parallel lines are all the same distance apart.). The point X trisects the segment AB. As TR pointed out, the method can be used to produce any proportion. In the bottom part of the same diagram, point X divides AB into pieces 7/12 and 5/12.

The above diagrams were drawn with the program Cinderella.

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