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The circumcircle of three collinear points is the line on which the three points lie, often referred to as a circle of infinite radius. Nearly collinear points often lead to numerical instability in computation of the circumcircle. Circumcircles of triangles have an intimate relationship with the Delaunay triangulation of a set of points.
Any line through a triangle that splits both the triangle's area and its perimeter in half goes through the triangle's incenter (the center of its incircle). There are either one, two, or three of these for any given triangle. [15] The incircle radius is no greater than one-ninth the sum of the altitudes. [16]: 289
The radius of a triangle's circumcircle is twice the radius of that triangle's nine-point circle. [6]: p.153 Figure 3. A nine-point circle bisects a line segment going from the corresponding triangle's orthocenter to any point on its circumcircle. Figure 4
All triangles have a circumcircle, but not all quadrilaterals do. An example of a quadrilateral that cannot be cyclic is a non-square rhombus . The section characterizations below states what necessary and sufficient conditions a quadrilateral must satisfy to have a circumcircle.
Circumcircle, the circumscribed circle of a triangle, which always exists for a given triangle. Cyclic polygon, a general polygon that can be circumscribed by a circle. The vertices of this polygon are concyclic points. All triangles are cyclic polygons. Cyclic quadrilateral, a special case of a cyclic polygon.
To draw the circumcircle, draw two perpendicular bisectors p 1, p 2 on the sides of the bicentric quadrilateral a respectively b. The perpendicular bisectors p 1, p 2 intersect in the centre O of the circumcircle C R with the distance x to the centre I of the incircle C r. The circumcircle can be drawn around the centre O.