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Circumcenter Theorem


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.


Δ A B C , the perpendicular bisectors of A B ¯ , B C ¯ and A C ¯ .

To prove:

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

The perpendicular bisectors of A C ¯ and B C ¯ intersect at point O .

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

Draw O A ¯ , O B ¯ and O C ¯ .

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

So, O A = O C and O C = O B .

By the transitive property,

O A = O B .

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

So, O is on the perpendicular bisector of A B ¯ .

Since O A = O B = O C , 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 .