I was at brunch this morning and I met a young man of about 35 years who is a musician but studied mathematics in college. He mentioned to me that one of the great problems of mathematics is the problem of trisecting the angle. He taught me how to bisect an angle, and kept emphasizing that this pertained to Euclidean geometry. His description was that "in Euclidean geometry, you get a straight edge and a compass." How would one be able to prove that an angle has been bisected using one these tools and without applying theorems? I figured there must be some theorems to go along with this description. I want to be clear. What is Euclidean geometry? What does it include? What kind of theorems would Euclidean geometry include? I thought Euclid worked with Cartesian planes. Would it be valid to use concepts like slope in Euclidean geometry? I recognize this question is large in scope. Ideally, I would like to be refered to some book that could give me an in depth understanding of Euclid's work, but I would also like a quick description if you can.