Mazurkiewicz's Theorem on 2-Sets

Submitted by Sam Northshield, 03 June 2000. Original answer and this article by Allen Stenger.

Does there exist a subset of the plane such that every straight line intersects the set at exactly two points?

(Remark. A set that intersects every line of the plane in exactly n points is called an n-set. A 1-set is clearly impossible; this question asks whether 2-sets exist.)

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