Any three noncollinear points in a plane can be shown to be equidistant from some point in the plane they determine. Also three noncollinear points determine a triangle.
To begin lets start with a triangle.
Next construct the perdendicular bisectors of each side of triangle ABC, and lable the intersection P.
Next construct a segment from P to any vertex. Using P as the center and the segments as the radius construct a circle.
Clearly the circle passes through points A, B, and C. Therefore P is the point in the plane determined by the points A,B,& C that is equidstant from all three points. The name of the point P for triangle ABC is the Circumcenter. To see the proof that the 3 perpendicular bisects are concurrent click here.