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.

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

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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 three 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
Centroid		Located at intersection of medians See Triangle centroid definition

Equilateral triangle centers

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

Related triangle topics

General

Triangle types

Triangle centers

Congruence and Similarity

Triangle quizzes and exercises