Down, Over, Up

If you start at the North Pole and walk one mile south, one mile east, and one mile north, you’ll find yourself back at your starting point.

The North Pole is not the only point with this property on Earth’s surface. In fact, there are any number of such points. Where are they?

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There’s a ring of such points around the South Pole — from any one of them you’d walk south, march in a circle around the pole, and return north to your starting point. (Further solutions involve circling the pole multiple times.)