Through any three points that don’t sit on a line, there is EXACTLY ONE circle. Its center is the one place equidistant from all three — found where the perpendicular bisectors of the sides cross. It’s how you locate a lightning strike or a phone from three towers. Slide a point and watch the circle re-fit.
The circumcircle passes through all three vertices; its center (the circumcenter) is equidistant from them, which is why it lies at the intersection of the three sides’ perpendicular bisectors. It is computed here in closed form from the coordinates. A fail-loud self-check throws unless the distance from the circumcenter to each of the three points is IDENTICAL — the property that makes three range-or-bearing fixes pin a single location (the basis of trilateration).
The circumcenter can fall OUTSIDE an obtuse triangle (and to infinity as the points near collinear) — three near-aligned references give a weak fix, the real geometry behind ‘dilution of precision’. The equidistance and the closed-form center are exact.