8th Croatian Mathematical Congress

Minimum and maximum area of bicentric quadrilateral, hexagon and octagon

3 Jul 2024, 09:20

30m

D1 (Faculty of Economics and Business, J. J. Strossmayer University of Osijek)

Poster G: Geometry Poster session

Speaker

Mandi Orlić Bachler (Zagreb University of Applied Sciences)

Description

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.