The Circumcentre of a Triangle

The Circumcentre

Just one circle passes through three given points that are not in a straight line. It is the circumcircle of the triangle of points and its centre is the circumcentre, O, of the triangle. The circumcentre is the point of concurrency of the three perpendicular bisectors (also termed the mediators) of the sides of the triangle (since the perpendicular bisector of any chord of a circle passes through the centre of the circle, by symmetry).

The angle 'subtended' by a chord at any point on the circle, on the same side of the chord as the centre, is the same. This angle is equal to half the angle subtended at the centre. (Proof: We have angles A = V+W, B = W+U, C = U+V. So U+V+W = 90°. Angle subtended at centre by side BC is 2.(90°−U). Angle subtended at A by side BC is V+W = 90°−U.)

The circumcentre is internal in the case of an acute angled triangle, coincides with the mid-point of the longest side in the case of a right angled triangle, and is external to an obtuse-angle triangle, being in the area 'below' the longest side.

The circumcentre is the orthocentre of the triangle LMN formed by the mid-points of the sides, since LM, MN, NL are parallel to AB, BC, CA, since ANM and ABC and similar pairs are similar triangles.

The radius R of the circumcircle is R = (a.sinU)/2 = (b.sinV)/2 = (c.sinW)/2.

The angle in a semicircle is a right angle. The angle subtended at a point on the minor arc is 180° minus the angle subtended on the major arc. The opposite angles of a cyclic quadrilateral (i.e. one inscribed in a circle) add to 180°.