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Concept
Concyclic points
Summary
In geometry, a set of points are said to be concyclic (or cocyclic) if they lie on a common circle. A polygon whose vertices are concyclic is called a cyclic polygon, and the circle is called its circumscribing circle or circumcircle. All concyclic points are equidistant from the center of the circle.
Three points in the plane that do not all fall on a straight line are concyclic, so every triangle is a cyclic polygon, with a well-defined circumcircle. However, four or more points in the plane are not necessarily concyclic. After triangles, the special case of cyclic quadrilaterals has been most extensively studied.
In general the centre O of a circle on which points P and Q lie must be such that OP and OQ are equal distances. Therefore O must lie on the perpendicular bisector of the line segment PQ. For n distinct points there are n(n − 1)/2 bisectors, and the concyclic condition is that they all meet in a single point, the centre O.
Circumcircle
The vertices of every triangle fall on a circle called the circumcircle. (Because of this, some authors define "concyclic" only in the context of four or more points on a circle.) Several other sets of points defined from a triangle are also concyclic, with different circles; see Nine-point circle and Lester's theorem.
The radius of the circle on which lie a set of points is, by definition, the radius of the circumcircle of any triangle with vertices at any three of those points. If the pairwise distances among three of the points are a, b, and c, then the circle's radius is
The equation of the circumcircle of a triangle, and expressions for the radius and the coordinates of the circle's center, in terms of the Cartesian coordinates of the vertices are given here and here.
In any triangle all of the following nine points are concyclic on what is called the nine-point circle: the midpoints of the three edges, the feet of the three altitudes, and the points halfway between the orthocenter and each of the three vertices.
Official source
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