If you do geometry (or approximation algorithms, for that matter) long enough, it becomes second nature to flip between primal and dual space. Points become hyperplanes, convex hulls become envelopes, and so on.
Here's a question that has been puzzling me: What is a good example of a problem where flipping between primal and dual (or even just going from one to the other) is the key to its solution ? Now obviously, there are NP-hard problems that admit a primal-dual solution, where by definition we need the dual space. I don't mean these kinds of problems.
I'm merely wondering if there is a relatively simple problem with a relatively simple solution that would be hard to explain without duality. The easiest that comes to mind is how the complexity of the convex hull in 3D is linear (because the complexity of the lower envelope in 3D is linear). Another potential example is the rotating calipers method for finding diameter/width etc: it can be much easier to see what's going on in the dual space.