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:

ΔABC , the perpendicular bisectors of AB ¯ , BC ¯ and AC ¯ .

To prove:

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

The perpendicular bisectors of AC ¯ and BC ¯ intersect at point O .

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

Draw OA ¯ , OB ¯ and OC ¯ .

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 AB ¯ .

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.