What is the circumcenter?

How the outside circle is built

The circumcenter is equidistant from all three vertices because any point on a perpendicular bisector is equally distant from the endpoints of that side. Where the bisectors meet, that equality holds for the whole triangle.

Its location depends on the triangle type: inside an acute triangle, at the midpoint of the hypotenuse in a right triangle, and outside an obtuse triangle. That makes this center especially useful for comparison.

To find it, draw perpendicular bisectors of two sides. Their intersection is enough because a point on a perpendicular bisector is equally far from the two endpoints. The third side's bisector should pass through the same point if the construction is accurate.

In a right triangle, the circumcenter lands at the midpoint of the hypotenuse. The three vertices then sit on a circle whose diameter is that hypotenuse, which connects the triangle-center idea back to the circle theorem about angles in a semicircle.

• The circumcenter is the point where the perpendicular bisectors intersect.
• The circumcenter is created by the three perpendicular bisectors of the sides.
• It is the center of the circumcircle and is equally distant from the three vertices.
• Its position changes with triangle type: inside acute, on the hypotenuse midpoint in right, outside obtuse.

Where the circumcenter helps

• Use the circumcenter when constructing or analysing a triangle's circumcircle.
• Use it in proofs that depend on equal distances from one point to all three vertices.
• Use it in right-triangle problems involving the midpoint of the hypotenuse.

Three circumcenter checks

• Do not confuse perpendicular bisectors of sides with altitudes or medians.
• Do not assume the circumcenter must lie inside the triangle.
• Do not confuse equal distance to vertices with equal distance to sides, which belongs to the incenter.