Final answer:
The circumcenter, defined as the point where the perpendicular bisectors of a triangle's sides intersect, is equidistant from the triangle's vertices, fulfilling the definition of a circle's center. This proves the circumcenter is the center of a circle circumscribing triangle ABC.
Step-by-step explanation:
To prove that the circumcenter is the center of a circle incident to the vertices of triangle ABC, we first note that the circumcenter is the point equidistant from all three vertices of the triangle. The definition of a circle is that all points on the circle are equidistant from the center. By the definition of a circumcenter, it is found at the intersection of the perpendicular bisectors of the sides of the triangle, which guarantees its equidistance to the vertices of the triangle, satisfying the definition of a circle's center.
The circumcenter being the point where each vertex of △ABC is thrice incident with the circle implies that the circle circumscribes the triangle. Any point on the perpendicular bisector of a segment is equidistant from the endpoints of that segment. Since all three perpendicular bisectors intersect at the circumcenter, it must be equidistant from all three vertices, proving it as the center of the circumscribed circle.