A bicentric quadrilateral, also called a cyclic-inscriptable quadrilateral, is a four-sided bicentric polygon . The inradius ,
circumradius , and offset are connected by the equation
(1)
(Davis; Durége 1861; Casey 1888, pp. 109-110; Johnson 1929; Dörie 1965; Coolidge 1971, p. 46; Salazar 2006). Finding this relation is sometimes known as Fuss's problem.
In addition
(Beyer 1987), where is the semiperimeter , and
(4)
The area of a bicentric quadrilateral is
where
and
are the lengths of the diagonals (Ivanoff 1960; Beyer 1987, p. 124).
See also Bicentric Polygon ,
Bicentric Triangle ,
Cyclic Quadrilateral ,
Poncelet's
Porism
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References Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 124,
1987. Bogomolny, A. "Easy Construction of Bicentric Quadrilateral."
http://www.cut-the-knot.org/Curriculum/Geometry/BicentricQuadri.shtml . Bogomolny,
A. "Easy Construction of Bicentric Quadrilateral II." http://www.cut-the-knot.org/Curriculum/Geometry/BicentricQuadri2.shtml . Casey,
J. A
Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction
to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges,
Figgis, & Co., 1888. Coolidge, J. L. A
Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, 1971. Davis,
M. A. Educ. Times 32 . 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. Durége, H. Theorie der
elliptischen Functionen: Versuch einer elementaren Darstellung. Leipzig, Germany:
Teubner, p. 185, 1861. Ivanoff, V. F. "Solution to Problem
E1376: Bretschneider's Formula." Amer. Math. Monthly 67 , 291-292,
1960. Johnson, R. A. Modern
Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle.
Boston, MA: Houghton Mifflin, pp. 91-96, 1929. Salazar, J. C.
"Fuss's Theorem." Math. Gaz. 90 , 306-308, 2006. Referenced
on Wolfram|Alpha Bicentric Quadrilateral
Cite this as:
Weisstein, Eric W. "Bicentric Quadrilateral."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/BicentricQuadrilateral.html
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