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Definition of the circumcenter of a triangle:

The circumcenter of a triangle in the point in the same plane as the triangle that is equidistant from the three vertices of the triangle. We know the perpendicular bisector of a segment is equidistant from the endpoints of the segment. Thus, the circumcenter of a triangle is where the perpendicular bisectors of the three sides of the traingle intersect. However, it remains to show all three perpendicular bisectors intersect.