So, you think that a given circle circumscribes at most one triangle?

Agree.

What a bad practice of confusing the area A with the set A on the part of the problem authors. B should be the set of (nonnegative) real numbers. The area of a triangle is an individual number, it can't qualify for a set B.

Judging by the curly braces, {p, q} is a set of points and not an ordered pair. B should be a set of sets with at most two elements. Why do you think this is not a function?