The Circumcenter of a triangle
Latin: circum - "around" centrum - "center"
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
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.
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
- 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.
Related triangle topics
Perimeter / Area
Congruence and Similarity
Triangle quizzes and exercises
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