Bicentric Quadrilateral
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
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. http://mathworld.wolfram.com/BicentricQuadrilateral.html
Contact the MathWorld Team
quadrilaterals
