Bicentric Polygon

$\begin{figure}\begin{center}\BoxedEPSF{BicentricQuadrilateral.epsf}\end{center}\end{figure}$

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

$\begin{displaymath} R^2-s^2=2Rr, \end{displaymath}$

(1)

where

is the Circumradius,

is the Inradius, and

is the separation of centers. In 1798, N. Fuss characterized bicentric Polygons of

, 5, 6, 7, and 8 sides. For bicentric Quadrilaterals (Fuss's Problem), the Circles satisfy

$\begin{displaymath} 2r^2(R^2-s^2)=(R^2-s^2)^2-4r^2s^2 \end{displaymath}$

(2)

(Dörrie 1965) and

$\displaystyle r$	$\textstyle =$	$\displaystyle {\sqrt{abcd}\over s}$	(3)
$\displaystyle R$	$\textstyle =$	$\displaystyle {1\over 4}\sqrt{(ac+bd)(ad+bc)(ab+cd)\over abcd}$	(4)

(Beyer 1987). In addition,

$\begin{displaymath} {1\over(R-s)^2}+{1\over(R+s)^2}={1\over r^2} \end{displaymath}$

(5)

and

$\begin{displaymath} a+c=b+d. \end{displaymath}$

(6)

The Area of a bicentric quadrilateral is

$\begin{displaymath} A=\sqrt{abcd}. \end{displaymath}$

(7)

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.

See also Poncelet's Closure Theorem

References

Beyer, W. H. (Ed.) CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 124, 1987.

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.