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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

Delta_m=sum_(k=0)^(m-1)(m-k)(2m+1; k)
(1)
=1/2[(2m+1)(2m; m)-2^(2m)]
(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, ... (OEIS A000531).

For triangles (n=3=2·1+1), the polynomial is Heron's formula, which may be written

 16K^2=2a^2b^2+2a^2c^2+2b^2c^2-a^4-b^4-c^4,
(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

16K^2=-a^4+2a^2b^2-b^4+2a^2c^2+2b^2c^2-c^4+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, Japanese Theorem

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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 Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Cyclic Polygon

Cite this as:

Weisstein, Eric W. "Cyclic Polygon." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CyclicPolygon.html

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