TOPICS
Search

Bicentric Polygon


BicentricPolygons

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

 R^2-x^2=2Rr,
(1)

where R is the circumradius, r is the inradius, and x is the separation of centers. For bicentric quadrilaterals, a result sometimes known as Fuss's problem, the circles satisfy

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

(Dörrie 1965, Salazar 2006) or, in another form,

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