Circumcenter

The circumcenter is the center of the circumcircle of a triangle—the unique circle that passes through all three vertices of a nondegenerate triangle. It is the point equidistant from the triangle’s vertices, and the common distance is called the circumradius.

Construction and definition: The circumcenter is the intersection point of the perpendicular bisectors of any two sides of the triangle. Equivalently, given three noncollinear points, there exists a unique circle through them, and its center is the circumcenter.

Location in relation to the triangle: In an acute triangle, the circumcenter lies inside the triangle; in a right triangle it lies at the midpoint of the hypotenuse; in an obtuse triangle it lies outside the triangle.

Properties and formulas: The circumradius R satisfies R = abc/(4Δ), where a, b, and c are the side lengths and Δ is the area of the triangle. If the vertices have coordinates (x1,y1), (x2,y2), and (x3,y3), the circumcenter can be found by solving the equations of the perpendicular bisectors or by using a determinant-based formula. The circumcenter is commonly denoted O and is one of the classic triangle centers (often listed as X(3) in triangle-center catalogs).

Notes and general context: The circumcenter is the center of the circle that passes through all three vertices. For a set of four or more points, a single circumcenter exists only if all points are concyclic; otherwise no common circumcircle passes through them.