for any triangle whatsoever, there is exactly one circle containing all three vertices of the triangle. ("sketch of proof". the locus of points equidistant from two given points is a straight line that is called the perpendicular bisector of the line segment connecting the points. the perpendicular bisectors of any two sides of a triangle intersect in exactly one point. this point must be equidistant from the vertices of the triangle.) this circle is called the circumcircle of the triangle.