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This question is a follow up of my previous one (Planar sets closed under intersection of circles, Planar sets closed under intersection of circles) and is motivated by G. Zaimi's answer https://...

Let $P$ be the plane with a point at infinity. By plane, I mean the Euclidian plane, and therefore it has circles. A line is also a circle, though its center is at infinity. If $A\subset P$ has ...