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how can you divide a line to three equal parts only by using a pair of compasses?
honestly, i don't think it can be done by using 'a pair of compasses' only, since one would probably need to use a straightedge as well.
...BUT,
i know an easier 'unpublished' method using a triangle (although i oftentimes do with a ruler). i've discovered it on my own and used it instead. i call my method: odd-sectioning ...as i can divide a regular figure (including lines of course, but i usually apply it on rectangles) into 3, 5, 7, 9... sections. i wish i could teach you, but i don't know how that is possible through here.
i also tried the method on perspectives, and it works similarly. i wouldn't wonder if you won't believe me because i'm sure it's not found in textbooks as i never got around to publish it -i got into other (more girly) things. but if you seek new knowledge, perhaps i can try sharing.
ADDED/EDITED:
wow, thumbs-down: ok let me try (this is not using the compass and straightedge, as shown by their links)
it's pretty simple, but requires a little analytical skill. i presume you know how to divide a rectangle into two parts, first by drawing diagonals (inside the figure) and dividing it along the intersection made by the diagonals. this is the way i analyzed it:
http://www.flickr.com/photos/31105109@N03/3008087426/
observation 1.
i divided the rectangle (above) into two (2)sections by using '2' diagonal lines -thereby producing an intersection thru which i can project a line (bisector). therefore... if i want to divide it into three (3) portions, i have to use '3' diagonal lines, and come up with two intersections (hence, i will have two lines/sectors to divide the figure into three)
observation 2.
i divided the rectangle (a single figure) into two, therefore if i have to divide a figure into three, i should first make the figure appear as 'two' identical figures (set side by side).
execution:
after some trial and error on paper (i'm more graphic than numerical), i came up with the three diagonals, two diagonals from the resulting halves, and one from the figure as a whole -so constructed as to intersect the other two (parallel) diagonals.
http://www.flickr.com/photos/31105109@N03/3008087432/
(perhaps you can try it on paper) the three diagonals produced two intersections, and after passing sectors thru each intersection. there it was -the rectangle divided in three parts. of course i had to check it afterwards using mathematical calculations and it stood correct. i later found out also, that by starting with dividing the figure into 4 sections -i can divide it thereafter by five.
pls try, and give me your feedback. i've done some informal lectures on this of course... i helped some friends taking up civil-engineering before, using this method to easily locate L/4 and L/5 points (critical in analyzing beams they say)