how to find the circumcenter of a triangle

The Circumcenter of a triangle

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. 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.