Here is a solution I found for dividing a line in 3 (or any other number of divisions) using a compass and a straight edge.

First construct a line at right angles to the line you want to divide, use the compass to make the line the same length as the first line. Then use the compass to make 2 marks along the second line, and one mark along the first line. Drawing a random circle off to the right is completely optional.

Draw a line to construct a right triangle with the ratio 3:1 between it’s two sides, then construct a line perpendicular to the hypotenuse by drawing two circles centered on the line.

Make another perpendicular line which is perpendicular to the previous one, but also passes through the end of second line we drew. This line will divide the original line, passing through at a third of the distance along.

Either repeat steps to divide the remaining part, set compass to the length of 1/3 we know have found or bisect the remaining portion of line in this case.

This a much easier problem than dividing an angle into 3 equal parts.

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Filed under: Math | Tagged: geometry, mathematics, Maths |

Rob Oxspring, on 25 April, 2008 at 6:43 pm said:Admittedly I tired after a grueling week at work but I can’t help feeling that a diagram would help this posting.

Hope all’s well,

Rob

mathew, on 25 April, 2008 at 7:18 pm said:Hmm did they not load? It pretty much requires on you being able to see the 3 I put. I think I even put `alt’ tags.

roxspring, on 26 April, 2008 at 2:28 pm said:Apparently the diagrams load fine if you’re reading from the original source. I was reading through google reader, which apparently didn’t bring the images through, and was very dry ðŸ™‚

At least now I have half an idea what you’re on about!

Rich, on 15 May, 2008 at 12:34 pm said:Ensure the first line’s at right angles using a compass and straightedge is possible, but it’s not trivial…

Mathew, on 15 May, 2008 at 8:25 pm said:I think it is trivial, I asked a friend and he said so too ðŸ™‚

He also showed me a simpler solution to the problem above.

Tenaciously, on 19 June, 2008 at 11:09 am said:Somehow i missed the point. Probably lost in translation ðŸ™‚ Anyway … nice blog to visit.

cheers, Tenaciously.