Two curves are consistent if it is possible for some point to be on both. (Being on one curve is consistent with being on the other.) There is an intersection. (Possibly many intersections.)
Two curves are inconsistent is it is impossible for any point to be on both. (Being on one curve is inconsistent with being on the other — it contradicts, being on the other.) There is no intersection.
Statements are consistent if it is possible for both to be true, statements are inconsistent if it is not possible for both to be true. (The truth of one is consistent or inconsistent with the truth of the other.)
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