Triangle Circumcenter definition - Math Open Reference

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The Circumcenter of a triangle

Latin: circum - "around" centrum - "center"

The point where the three perpendicular bisectors of a triangle meet.
One of a triangle's points of concurrency.

Try this Drag the orange dots on each vertex to reshape the triangle. Note the way the three perpendicular bisectors always meet at a point - the circumcenter

One of several centers the triangle can have, the circumcenter is the point where the perpendicular bisectors of a triangle intersect. The circumcenter is also the center of the triangle's circumcircle - the circle that passes through all three of the triangle's vertices. As you reshape the triangle above, notice that the circumcenter may lie outside the triangle.

Special case - right triangles

In the special case of a right triangle, the circumcenter (C in the figure at right) lies exactly at the midpoint of the hypotenuse (longest side). See also Circumcircle of a triangle.

Finding the circumcenter

It is possible to find the circumcenter of a triangle using construction techniques using a compass and straightedge. See Construction of the Circumcircle of a Triangle has an animated demonstration of the technique, and a worksheet to try it yourself. The circumcenter is found as a step to constructing the circumcircle.

Summary of triangle centers

There are many types of triangle centers. Below are four of the most common.

Incenter		Located at intersection of the angle bisectors. See Triangle incenter definition
Circumcenter		Located at intersection of the perpendicular bisectors of the sides. See Triangle circumcenter definition
Centroid		Located at intersection of medians. See Centroid of a triangle
Orthocenter		Located at intersection of the altitudes of the triangle. See Orthocenter of a triangle

In the case of an equilateral triangle, all four of the above centers occur at the same point.

The Euler line - an interesting fact

It turns out that the orthocenter, centroid, and circumcenter of any triangle are collinear - that is, they always lie on the same straight line called the Euler line, named after its discoverer.
For more, and an interactive demonstration see Euler line definition.

Summary of triangle centers

The Euler line - an interesting fact

Related triangle topics

General

Perimeter / Area

Triangle types

Triangle centers

Congruence and Similarity

Triangle quizzes and exercises