GeoProof — Circumcenter & Circumcircle explorer

The circumcenter is equidistant from all three vertices — that distance is the circumradius R

The Circumcenter & Circumcircle

The perpendicular bisector of a segment is the line of all points equally far from its two endpoints. Draw the perpendicular bisector of each side of a triangle and all three meet at one point — the circumcenter O:

three perpendicular bisectors meet at the circumcenter O

Because O lies on all three bisectors, it is the same distance from all three vertices. That distance is the circumradius R, and the circle centered at O through A, B, C is the circumcircle.

Why equidistant?

A point on the perpendicular bisector of BC is equally far from B and C. A point on all three bisectors is equally far from A, B and C at once — so OA = OB = OC.

Inside, on, or outside

For an acute triangle the circumcenter is inside; for a right triangle it lands exactly on the midpoint of the hypotenuse (so the hypotenuse is a diameter); for an obtuse triangle it falls outside the triangle. Drag a vertex to stretch the triangle obtuse and watch O leave.

A useful formula

The circumradius relates to the sides and area by R = abc ÷ (4·Area), and to the law of sines by a ÷ sin A = 2R.

Where it's used

The circumcircle underlies the inscribed-angle theorem, the law of sines, the nine-point circle, and the Euler line (O is one of its three named centers).