Circumcircle

The circumcircle of a triangle is the unique circle passing through all three vertices. Its center is the triangle’s circumcenter.

Triangle: A polygon with three sides and three vertices. Its three vertices determine the circumcircle.
Circumradius: The radius of a triangle’s circumcircle, equal to the distance from its circumcenter to any vertex. It specifies the size of the circle through the three vertices.
Cyclic quadrilateral: A quadrilateral whose four vertices lie on a single circle. A triangle’s circumcircle can contain a fourth point, forming a cyclic quadrilateral.
Incircle: The largest circle contained in a triangle, tangent to all three sides. Unlike the circumcircle, it touches the sides rather than passing through the vertices.
Perpendicular bisector: A line that meets a segment at its midpoint at a right angle. The perpendicular bisectors of triangle sides meet at the circumcenter.
Law of sines: A triangle identity relating each side length to the sine of its opposite angle. It gives the circumradius as a side divided by twice its opposite angle’s sine.
Miquel point: The common point of three circles associated with the sides of a complete quadrilateral. Miquel’s theorem uses circles through triangle vertices and points on its sides.
Excircle: A circle tangent to one side of a triangle and the extensions of its other two sides. It is defined by side tangencies, unlike the circumcircle’s vertex condition.
Circumcenter: The point equidistant from a triangle’s three vertices and the center of its circumcircle. It is the center from which the circumcircle’s radius is measured.
Extended law of sines: The identity stating that a triangle’s side lengths divided by their opposite angle sines equal twice its circumradius. It connects side lengths and angles directly to the circumcircle’s radius.

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