[upperhalfplane]

My favorite geometric proof of an inequality is the one I read on Terry Tao's blog. Interestingly, it's not presented as a geometric proof, but it is very much one: if you have two vectors x, y, you just shrink the longer one and grow the shorter one until they reach the same size, without changing the LHS and the RHS of the inequality. Then you expand the norms of ||x - y||^2>=0 and ||x + y||^2>=0 and see -||x||^2 - ||y||^2 <= 2<x,y> <= ||x||^2 + ||y||^2, and since ||x||=||y|| you get the result.