“Cyclic Polygon”的概念、定义、习题解题方法及翻译-数学辞典

单词

Cyclic Polygon

释义

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)

and which is of order

. For a Cyclic Quadrilateral, the Polynomial isBrahmagupta's Formula, which may be written


			(4)

which is of order

. 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.

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