# Cyclic Polygons

## Cyclicity

A circumcyclic polygon has an in-circle touching all its sides. An incyclic polygon has a circle through all its vertices. Either of these polygons can be called cyclic polygons. If either circle exists it is uniquely defined. A polygon with incircle and circumcircle is bicyclic. All triangles are bicyclic. All regular polygons are bicyclic. An acyclic polygon has neither incircle nor circumcircle.

To find the circumcentre of an incyclic polygon: draw perpendicular bisectors (mediators) of its sides; these all meet at the circumcentre. To find the incentre of a circumcyclic polygon draw bisectors of the internal angles; these all meet at the incentre.

Given two circles, one inside the other, from any point on the outer circle we can draw a tangent to the inner circle which will meet the outer circle again. From this new point on the outer circle we can draw a further tangent to the inner circle, meeting the outer circle at a third new point. We can continue this process, thus generating a cyclic open polygon. If we continue the process long enough the polygon will close (or will do so to within a degree of accuracy).