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The bicentric 𝑛-gon is a polygon with 𝑛 sides that are tangential for incircle and chordal for circumcircle. The connection between the radius (𝑅) of circumcirle, the radius (𝑟) of incircle and the distance (𝑑) of their centers represents the Fuss' relation for the certain bicentric 𝑛-gon. According to Poncelet's porism, there exist infinitely many bicentric 𝑛-gons with these characteristics. In this presentation, we derive relations for the largest and smallest area as a function of the parameters 𝑅, 𝑟 and 𝑑 for the bicentric quadrilateral, hexagon and octagon. We used the works [1]-[2] by Josefson to find out which 𝑛-gons have extreme areas for bicentric quadrilaterals, and work [3] by Radić for the bicentric hexagon and octagon. So far, only the relation for calculating the area of a bicentric quadrilateral is known. The corresponding relations are derived in [1], [2], [4].
References:
[1] M. Josefsson, Maximal Area of a Bicentric Quadrilateral, Forum Geometricorum, Volume 12 (2012) 237-241.
[2] M. Josefsson, Minimal area of a bicentric quadrilateral, The Mathematical Gazette, 99(545), 237-242, 2015.
[3] M. Radić, Certain inequalities concerning bicentric quadrilaterals, hexagons and octagons, Journal of Inequalities in Pure and Applied Mathematics 6 (2005)
[4] M. Orlić Bachler, Z. Kaliman, O površini bicentričnog četverokuta, Acta mathematica Spalatensia. Series didactica 6, 99-114, 2023.
area of bicentric polygon; quadrilateral; hexagon; octagon
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