It has become obvious that the Shapes category was in way over its head with all sorts of geometrical problems. So, the hard-core Geometry problems will now reside here, where they will fit in much better.
There is this old chestnut: "A man walks a mile South, a mile East, and a mile North and ends up right back where he started. Where did he start?"
Aside from the timeworn singular point answer, give another answer that includes a countably infinite number of sets, each containing an uncountably infinite number of points that satisfy the problem. (Assume a smooth spherical globe, and no tricks.)