Cyclic Polygon
A cyclic polygon is a Polygon with Vertices upon which a Circle can be Circumscribed. Since everyTriangle has a Circumcircle, every Triangle is cyclic. It is conjectured that for a cyclic polygonof 
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 twoPolynomials 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) |
in . For a Cyclic Quadrilateral, the Polynomial isBrahmagupta's Formula, which may be written | |  | |
 |  | (4) |
which is of order
in . Robbins (1995) gives the corresponding Formulas for theCyclic Pentagon and Cyclic Hexagon.See also Concyclic, Cyclic Hexagon, Cyclic Pentagon, Cyclic Quadrangle, Cyclic Quadrilateral
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 A000531in ``The On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html.
- Haberdasher's Problem
- Hadamard Design
- Hadamard Matrix
- Hadamard's Inequality
- Hadamard's Theorem
- Hadamard Transform
- Hadamard-Vallée Poussin Constants
- Hadwiger Problem
- Hadwiger's Principal Theorem
- Hafner-Sarnak-McCurley Constant
- Hahn-Banach Theorem
- Hailstone Number
- Hairy Ball Theorem
- Half
- Half-Closed Interval
- Half-Normal Distribution
- Half-Open Interval
- Halley's Irrational Formula
- Halley's Method
- Halley's Rational Formula
- Hall-Janko Group
- Halphen Constant
- Halphen's Transformation
- Halting Problem
- Hamiltonian Circuit