A polygon which has both a circumcircle (which touches each vertex) and an incircle (which is
tangent to each side). All triangles are bicentric with
 |
(1)
|
where 
is the circumradius, 
is the inradius, and 
is the separation of centers. For bicentric
quadrilaterals, a result sometimes known as Fuss's problem, the circles
satisfy
 |
(2)
|
(Dörrie 1965, Salazar 2006) or, in another form,
 |
(3)
|
(Davis; Durége 1861; Casey 1888, pp. 109-110; Johnson 1929; Dörrie 1965).
If the circles permit successive tangents around the incircle which close the polygon for one starting point on the
circumcircle, then they do so for all points on
the circumcircle, a result known as Poncelet's
porism.
See also
Bicentric Quadrilateral,
Bicentric Triangle,
Circumcircle,
Incircle,
Polygon,
Poncelet's
Porism,
Poncelet Transverse,
Tangential
Quadrilateral,
Triangle,
Weill's
Theorem
Explore with Wolfram|Alpha
References
Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 124,
1987.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.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.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 Polygon
Cite this as:
Weisstein, Eric W. "Bicentric Polygon."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BicentricPolygon.html
Subject classifications
