Circumcentre of a Triangle

The circumcentre of a triangle is the point where the three perpendicular bisectors of the triangle meet. It is the centre of the triangle’s circumcircle — the circle that passes through all three vertices of the triangle.

            A
           / \
      O---/   \
         /     \
        B-------C
   O = circumcentre

Definition: The circumcentre of a triangle is the common point where the three perpendicular bisectors of its sides intersect.

If perpendicular bisectors of \(AB\), \(BC\), and \(CA\) meet at point \(O\), then \(O\) is the circumcentre of \(\triangle ABC\).

• The circumcentre is equidistant from all three vertices.
• It is the centre of the circumcircle that passes through vertices \(A\), \(B\), and \(C\).
• Its position (inside, on, or outside the triangle) depends on the type of triangle.
• The radius of the circumcircle is the distance from the circumcentre to any vertex.

The circumcentre changes position depending on the type of triangle:

The circumcircle of a triangle is the unique circle that passes through all three vertices of the triangle.

The circumcentre is the centre of this circle, and the radius is:

\( R = OA = OB = OC \)

         A
      o     o
    B    O     C
      o     o
   (circumcircle)

To construct the circumcentre of \(\triangle ABC\):

1. Construct the perpendicular bisector of side \(AB\).
2. Construct the perpendicular bisector of side \(BC\).
3. The point where they intersect is the circumcentre \(O\).
4. (The perpendicular bisector of \(CA\) will also pass through \(O\).)

The circumcentre is important because it helps in:

• constructing circumcircles around triangles,
• solving geometry problems involving vertex distances,
• designing stable triangular frames and structures,
• coordinate geometry applications.

The equal distance property makes the circumcentre especially valuable in geometric constructions and proofs.