The more I study algebraic geometry, the more I realize how I should have studied projective geometry in depth before. Not that I don't understand projective space (on the contrary, I am well versed in several different constructions of it), but I lack the familiarity with basic results as cross-ratios, how projective linear transformations act on projective space (as in how many points determine one transformation), Desargues' theorem, etc. I also sometimes feel that it wouldn't hurt to get more practice with hard (as in Olympiad-style) classical geometry problems that may or may not use some facts of projective geometry.
To summarize, I am looking for a reference that covers classical results of projective geometry, and yet assumes the maturity of a reader who has already started studying algebraic geometry. It would be only better if such a book could help me understand where those amazing solutions to Olympiad problems come from.
Does anyone have a suggestion?