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Bicentric Quadrilateral

BicentricQuadrilateral

A bicentric quadrilateral, also called a cyclic-inscriptable quadrilateral, is a four-sided bicentric polygon. The inradius r, circumradius R, and offset x are connected by the equation

1/((R-x)^2)+1/((R+x)^2)=1/(r^2)
(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

r=(sqrt(abcd))/s
(2)
R=1/4sqrt(((ac+bd)(ad+bc)(ab+cd))/(abcd))
(3)

(Beyer 1987), where s is the semiperimeter, and

a+c=b+d.
(4)

The area of a bicentric quadrilateral is

A=sqrt(abcd)
(5)
=1/2sqrt(p^2q^2-(ac-bd)^2),
(6)

where p and q 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.

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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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