I have three students in my secondary geometry class that just destroy everything I throw at them. I tasked them with writing the word problems for their midterms and one of the three wrote simply "What are the interior angle measures of an arch made of five isosceles trapezoids?" (I had to cheat to get the answer:) really astounding minds that I want to challenge with some tough geometric constructions because I think they would really get a lot out of the assignment. They really enjoy when we do constructions in class. What are some challenging geometric constructions that I can give to my students?
There are two websites that make geometric construction into a game. NiloCK mentioned euclidthegame.com in his comment. There is another construction game at sciencevsmagic.net, with a very different look and feel. I think some people will one enjoy one or the other much more, though I enjoy them both. I have only made it through about half the challenges on the second site.
Suggest posing these constructions: