Definition: The circumcenter of a triangle is the point where the three perpendicular bisectors of its sides meet; it is equidistant from all three vertices.
What does the circumcenter do?
Since it is the same distance from each vertex, it is the center of the circumscribed circle, the circle passing through all three vertices. That common distance is the circumradius.
Where is it located?
Inside an acute triangle, outside an obtuse triangle, and at the midpoint of the hypotenuse of a right triangle.
Example. A right triangle has vertices (0, 0), (10, 0), and (0, 24). The hypotenuse runs from (10, 0) to (0, 24), with midpoint (5, 12). So the circumcenter is (5, 12). The circumradius is half the hypotenuse. The hypotenuse is the square root of 100 + 576, which is 26, so the radius is 13. Check: the distance from (5, 12) to the origin is the square root of 25 + 144, which is 13.
What mistakes occur?
Using angle bisectors or medians by mistake. Only the perpendicular bisectors give this point. Also, remember a perpendicular bisector must pass through the midpoint of the side, not just be perpendicular to it.