once you say a "circle" is "a convex polygon with every point equidistant from a center,"

Yikes, bad definition. Better: "the set of all points a given distance from a center". (And, of course, such a set isn't going to be convex; you may be thinking of a disc .)

As for the relation of mathematics to "reality", the reason no "empirical confirmation" of mathematical truths is needed is because "mathematics" is defined that way: it's the subject in which you examine models internally, for their own sake. (Once upon a time, some people in ancient Greece decided it would be a good idea to have a subject like this; they seem to have been right.) If it turns out that mathematics always somehow manages to describe the world accurately....well, there's always going to be some model that works, isn't there? Mathematics is the study of models.