Circumcenter (O)

The circumcenter O is the intersection of the perpendicular bisectors of the sides, and it is the center of the circle passing through all three vertices.

Keywords: circumcenter, perpendicular bisector, circumcircle, equidistant, triangle centers

Prerequisites: triangle · Difficulty: beginner

The circumcenter is a special point that is the same distance from all three corners (vertices) of the triangle.

The Circumcircle

Because the circumcenter is equally far from each vertex, you can draw a circle around the triangle that touches all three vertices. This circle is called the circumcircle. The circumcenter is the exact center of this circle.

How do we find it?

To find the circumcenter, we draw the perpendicular bisectors of the triangle's sides.

• A bisector is a line that cuts a side exactly in half (at its midpoint).
• Perpendicular means the line makes a perfect right angle ($90°$) with the side.

Each triangle has three sides, so it has three perpendicular bisectors. They all meet at one point: the circumcenter!

Fun Fact

• For an acute triangle (all angles are less than $90°$), the circumcenter is inside the triangle.
• For a right triangle, the circumcenter is on the midpoint of the longest side (the hypotenuse).
• For an obtuse triangle (one angle is more than $90°$), the circumcenter is outside the triangle!