You win Moonray. In fact there are an infinite number of points on the "ring" where you can start from. In fact there are an infinite number of similar rings. The second one a little South of what you described where after you go South 1 mile, when you go east you go "around" the earth in an easterly direction two times as it is 0.5 miles around. Third sub ring is 1/3 mile around, etc. You may want to have a heavy fur coat and some snow shoes however! You got my upvote. Congratulations. I have another riddle coming up in a bit...keep tuned.
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Haha that was fun! Thanks :) I follow you for more interesting posts.