The circumcenter of a triangle is a fundamental concept in geometry. It is defined as the point where the perpendicular bisectors of all three sides intersect. This special point has the remarkable property of being equidistant from all three vertices of the triangle.
For an acute triangle, where all angles are less than ninety degrees, the circumcenter is located inside the triangle. The circumcenter remains equidistant from all three vertices, as shown by the equal radii to each vertex.
For a right triangle, where one angle equals exactly ninety degrees, the circumcenter has a special location. It is positioned at the midpoint of the hypotenuse, which is the longest side opposite the right angle. In this case, the hypotenuse becomes the diameter of the circumcircle.
For an obtuse triangle, where one angle is greater than ninety degrees, the circumcenter is located outside the triangle. Despite being outside, the circumcenter maintains its fundamental property of being equidistant from all three vertices, as demonstrated by the equal radii of the circumcircle.
To summarize what we have learned about the circumcenter: It is the intersection point of perpendicular bisectors, always equidistant from vertices. Its location depends on triangle type - inside for acute triangles, on the hypotenuse for right triangles, and outside for obtuse triangles. The circumcenter serves as the center of the circumscribed circle that passes through all three vertices.