“Bicentric Polygon”的概念、定义、习题解题方法及翻译-数学辞典

单词

Bicentric Polygon

释义

Bicentric Polygon

A Polygon which has both a Circumcircle (which touches each vertex) and an Incircle (which is tangentto each side). All Triangles are bicentric with

(1)

where

is the Circumradius,

is the Inradius, and

is the separation of centers. In 1798, N. Fusscharacterized bicentric Polygons of

, 5, 6, 7, and 8 sides. For bicentricQuadrilaterals (Fuss's Problem), the Circles satisfy

(2)

(Dörrie 1965) and

			(3)
			(4)

(Beyer 1987). In addition,

(5)

and

(6)

The Area of a bicentric quadrilateral is

(7)

If the circles permit successive tangents around the Incircle which close the Polygon for one startingpoint on the Circumcircle, then they do so for all points on the Circumcircle.

See also Poncelet's Closure Theorem

References

Beyer, W. H. (Ed.) CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 124, 1987.

Dörrie, H. ``Fuss' Problem of the Chord-Tangent Quadrilateral.'' §39 in 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, pp. 188-193, 1965.

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