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This essential trains for: AMC-10, AMC-12, AIME, Math Kangaroo 9-10.

The circumcircle of a triangle is the circle that passes through all the vertices of the triangle. Since three non-collinear points in the plane determine a circle uniquely, there is only one circumcircle for any triangle. The center of the circumcircle (circumcenter) is found at the intersection of the perpendicular bisectors of the sides, since it has to be equidistant from the vertices. The radius of the circumcircle (circumradius) can be calculated as a function of the area and side lengths of the triangle.

Let the circle with center O be the circumcircle of scalene triangle ABC and let the side lengths be denoted with a, b, c as in the figure:

The radius of the circumcircle is traditionally denoted with R.

Angle A has measure θ and intercepts the minor arc BC. The central angle BOC intercepts the same minor arc and has, therefore, a measure equal to 2θ. Since the triangle BOC is isosceles (two of its sides are radii of the circle with center O), its altitide from O is also the angle bisector of the angle BOC. However, since the altitude is perpendicular to the chord BC, it bisects the chord into the congruent segments BM and MC:

We can use a sine ratio to relate the side length to the circumradius:

and use the formula for the area of the triangle if we know two side lengths and the angle between them:

Eliminate the trigonometric ratio between these expressions:

and solve for the circumradius: