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Circumcircles of Triangles

The circumcircle of a triangle is the unique circle on which all of its three vertices lie. The circumcenter of a triangle can be found as the intersection of the three perpendicular bisectors. (A perpendicular bisector is a line that forms a right angle with one of the triangle's sides and is erected from that side at its midpoint.) This is because the circumcenter is equidistant from any pair of the triangle's points. All points on the perpendicular bisectors are equidistant from the respective points of the triangle.

 

A triangle is acute (all angles smaller than a right angle) if the circumcenter lies inside the triangle. The triangle is obtuse (has an angle bigger than a right one) if the circumcenter lies outside the triangle. The triangle is a right triangle if the circumcenter lies on the hypotenuse. This is one form of Thales' theorem.

The diameter of the circumcircle can be computed as the length of any side of the triangle, divided by the sine of the opposite angle. (As a consequence of the law of sines, it doesn't matter which side is taken: the result will be the same.) The triangle's nine point circle has half the diameter of the circumcircle.

The circumcenter always lies on one line with the triangle's centroid and orthocenter. This line is known as Euler's line.