Intersection of any two polygons is implemented in Meshes.jl, and I highly recommend getting used to their types instead:
using Meshes
rect1 = Quadrangle((0.0,0.0), (1.0,0.0), (1.0,1.0), (0.0,1.0))
rect2 = Quadrangle((0.5,0.5), (1.5,0.5), (1.5,1.5), (0.5,1.0))
hasintersect(rect1, rect2)
We implemented the beautiful GJK algorithm:
This means that you can identify intersections between polygons, balls, or any geometry with well-defined support functions.
The case with rectangles could probably be optimized, and that is why we are always happy to review PRs from the community.
For anyone reading this thread in the future: most features of the packages used in the marked solution have been reimplemented in Meshes.jl. I do not recommend using these low-level packages unless you already have a large codebase built with them. You will end up reinventing the wheel all the time. Meshes.jl provides everything out of the box with cleaner (and sometimes faster) code.