Ten miles south, east, north and west

Question

I'm standing on the surface of the Earth. I walk ten miles south, ten miles east, ten miles north and ten miles west. I end up exactly where I started.
Where on earth am I?

sj2tpgk · Answer

I think the answer depends on what "going east/west" means. That is:

Under this definition, suppose you start from P1, walk south to P2, east to P3, north to P4 and west to P5. (Each walk is 10 miles long.)
I check the following three cases.
Case 1:

Case 2:

Case 3:

Bass · Answer

I think there are three(ish) choices. The simplest one is

To find the other solutions, we have to notice that

This means we can also start

and then we can, of course, start

Jeremy Dover · Answer

I think the answer is:

Reasoning:

Update: Based on @Bass's answer, I was curious about the actual calculation. Turns out it's really not that hard.

Ten miles south, east, north and west

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