Squaring the triangle with help

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] Yesterday, Numberphile released a delightful video with James Grime (redundant, I know) in which he walks us through a geometry problem first posed by Henry Dudeney in 1902. It’s known as the “Haberdasher’s Puzzle” and is reproduced in Dudeney’s 1907 book The Canterbury Puzzles. The problem was solved by C.W. McElroy after Dudeney published it in a periodical called the Weekly Dispatch. The problem is to cut up an equilateral triangle into pieces that can be translated and rotated (not flipped) to fit together into a square. The minimum number of pieces that allow you to do this is four, and here’s the arrangement shown in the video: What James doesn’t do in the video is explain how you position the cuts. Dudeney shows a straightedge-and-compass construction in his book, but I wanted an algebraic solution. This turned out to be more complicated…