So while not actually a specific problem I'm struggling with, I was hoping for some of your insight! For a course, I'm currently reading Stillwell's Four Pillars of Geometry. While it does a nice job of motivating the theory, I find it a little sparse with regard to challenging/engaging problems. Any suggestions as to where I can turn to find a nice supplementary text on projective/hyperbolic geometry?
There are several relatively recent textbooks on projective geometry and a host of pre-1950 texts. The most well known of the more recent ones is the probably the one by Coexeter. A little known book I consider a gem is Pierre Samuel's book.It's almost impossible to find now, but well worth tracking down for it's algebraic flavor. And of course, the major general axiomatic geometry texts-like Greenberg's classic and the more recent book by Hartshorne- have chapters on projective geometry.
Hyperbolic geometry is a lot harder to find a book on. I know there's a pretty good chapter in Greenberg, but other then that,I can't think of one off the top of my head.