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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 $2m+1$ sides, $16K^2$ (where $K$ is the Area) satisfies a Monic Polynomial of degree $\Delta_m$, where

$\displaystyle \Delta_m$ $\textstyle \equiv$ $\displaystyle \sum_{k=0}^{m-1} (m-k){2m+1\choose k}$ (1)
  $\textstyle =$ $\displaystyle {1\over 2}\left[{(2m+1){2m\choose m}-2^{2m}}\right]$ (2)

(Robbins 1995). It is also conjectured that a cyclic polygon with $2m+2$ sides satisfies one of two Polynomials of degree $\Delta_m$. The first few values of $\Delta_m$ are 1, 7, 38, 187, 874, ... (Sloane's A000531).


For Triangles $(n=3=2\cdot 1+1)$, the Polynomial is Heron's Formula, which may be written

\begin{displaymath}
16K^2=2a^2b^2+2a^2c^2+2b^2c^2-a^4-b^4-c^4,
\end{displaymath} (3)

and which is of order $\Delta_1=1$ in $16K^2$. For a Cyclic Quadrilateral, the Polynomial is Brahmagupta's Formula, which may be written
$\displaystyle 16K^2$ $\textstyle =$ $\displaystyle -a^4 + 2a^2b^2 - b^4 + 2a^2c^2 + 2b^2c^2 - c^4$  
  $\textstyle \phantom{=}$ $\displaystyle + 8abcd + 2a^2d^2 + 2b^2d^2 + 2c^2d^2 - d^4,$ (4)

which is of order $\Delta_1=1$ in $16K^2$. Robbins (1995) gives the corresponding Formulas for the Cyclic 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 A000531 in ``The On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.



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© 1996-9 Eric W. Weisstein
1999-05-25