Circumcenter Theorem

Circumcenter

The three perpendicular bisectors of a triangle meet in a single point, called the circumcenter.

Circumcenter Theorem

The vertices of a triangle are equidistant from the circumcenter.

Given:

, the perpendicular bisectors of

To prove:

The perpendicular bisectors intersect in a point and that point is equidistant from the vertices.

The perpendicular bisectors of intersect at point O.

Let us prove that point O lies on the perpendicular bisector of and it is equidistant from A, B, and C.

Draw

Any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment.

So, OA = OC and OC = OB.

By the transitive property,

OA = OB.

Any point equidistant from the end points of a segment lies on its perpendicular bisector..

So, O is on the perpendicular bisector of .

Since OA = OB = OC, point O is equidistant from A, B, and C.

This means that there is a circle having its center at the circumcenter and passing through all three vertices of the triangle.  This circle is called the circumcircle.