In HS geometry, our teacher started showing us basic constructions with compass and ruler: bisecting a line segment, drawing a line perpendicular to another line, and bisecting an angle. Then, at the end of class, he mentioned that no one had ever succeeded in figuring out a method to trisect angles. He did not mention that it had been proven that you couldn't (in the general case).

I spent the entire evening trying to trisect the angle.