How can I tell if two polygons intersect?

As Charlie already pointed out you can use the Separating Axis theorem. Check out this article for a C# implementation and example of polygon collision detection.

Strictly speaking, the other answers suggesting an algorithm are probably your best bet. But performance aside, you mentioned that you couldn't find anything like IntersectsWith for GraphicsPath or Region. However there is an Intersect method that updates a region to be the intersection of itself and another region or path.

You could create two regions, Intersect() one with the other, then test for Region.IsEmpty().

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