I was doing my usual blog-skimming this morning when I became very intrigued by this problem. Now, my calculus is admittedly rusty, but I think I've come up with a pretty simple geometric approach to the problem! I'm illustrating it here for the particular cases of n = 9 and n = 10. (Again, if you click on the pictures below, you can see enlarged versions in a separate window.)
Of course, I leave it to you to find out what the products would be. (Not difficult, obviously, considering all the lengths are already labeled for you.) What's getting me stuck is how to simplify this process into one neat formula, given any n, in order to prove the conjecture. (If you click through Sam Shah's post, you'll see at the bottom that I tried to formulate it at least in words, but my calculus fails me and I'm not sure how to write down one elegant formula for the whole thing!)
Anyway. I thought I'd share it with you, especially because GeoGebra helped me make such nice unit-circle diagrams, complete with color-coding! :)
Incidentally, I've been (true to my word) researching Geometry stuff for next year. In particular, Nancy Powell has some really lovely Geometry projects, that seem extremely rigorous and FUN! This was the first link that I came across when googling Geometry projects, so it's probably old news... but in case you haven't seen it, you MUST check it out!!
And, here is a cute SAT problem from David Marain -- this one is good for your middle-schoolers (and/or good for discussions about prime factorization)! :)