Concyclic points

In geometry, a set of points is said to be concyclic (or cocyclic) if they lie on a common circle.

Concyclic points, showing that the perpendicular bisectors of pairs are concurrent

Four concyclic points showing that angles α are the same. Points are the vertices of a cyclic quadrilateral

A circle can be drawn around any triangle. A quadrilateral that can be inscribed inside a circle is said to be a cyclic quadrilateral.

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 such lines to draw, and the concyclic condition is that they all meet in a single point.

A quadrilateral in which the four vertices are concyclic is called a cyclic quadrilateral. More generally, a polygon in which all vertices are concyclic is called a cyclic polygon. Three noncollinear points A, B, and C are concyclic to a single circle. Four different points A, B, C, and D are cyclic iff:

Four points in the complex plane are either concyclic or collinear if and only if their cross-ratio is real.

See also

• Collinear points
• Lester's theorem

External links

• Weisstein, Eric W., "Concyclic" from MathWorld.
• Four Concyclic Points by Michael Schreiber, The Wolfram Demonstrations Project.

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