I ran across this problem on page 90 in Mathematics in Historical Context by Jeff Suzuki: Find the length of the side of a equilateral pentagon inscribed in a square with one vertex coinciding with a corner of the square. Here is my solution.
I basically just pushed Pythagorean relations around the square to get a polynomial equation of degree four. My TI-84 Plus found an approximate zero. Abu Kamil found the exact answer in radical form. I wondered how he did that.
I had originally thought this might be good group problem for our Southern Oregon Math League high competition but it is too hard and doesn’t seem to lend itself to a group process. I however did pass the problem on to Dr. Curtis Feist. Here is his solution (my handwriting).
Now I saw how an exact solution could be found. I could relate Dr. Feist’s quadratic equation is related to my quartic equation this way.
His quadratic equation was of course a factor of my fourth degree equation. And with a little more trouble I found that my equation could be expressed as . In the spirit of Annalisa Crannell (see The Shad-Fack Transom) I wonder if this beautiful compact form reflects something fundamental about the geometry. The fourth power term looks daunting.