## Cyclic Polygon

A cyclic polygon is a Polygon with Vertices upon which a Circle can be Circumscribed. Since every Triangle has a Circumcircle, every Triangle is cyclic. It is conjectured that for a cyclic polygon of sides, (where is the Area) satisfies a Monic Polynomial of degree , where   (1)  (2)

(Robbins 1995). It is also conjectured that a cyclic polygon with sides satisfies one of two Polynomials of degree . The first few values of are 1, 7, 38, 187, 874, ... (Sloane's A000531).

For Triangles , the Polynomial is Heron's Formula, which may be written (3)

and which is of order in . For a Cyclic Quadrilateral, the Polynomial is Brahmagupta's Formula, which may be written     (4)

which is of order in . Robbins (1995) gives the corresponding Formulas for the Cyclic Pentagon and Cyclic Hexagon.

References

Robbins, D. P. Areas of Polygons Inscribed in a Circle.'' Discr. Comput. Geom. 12, 223-236, 1994.

Robbins, D. P. Areas of Polygons Inscribed in a Circle.'' Amer. Math. Monthly 102, 523-530, 1995.

Sloane, N. J. A. Sequence A000531 in The On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.