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Here we describe a general analytical derivation of Fuss' relation for bicentric polygons with an odd number of vertices ($n$-gons where $n geq 3$ is an odd integer). The bicentric $n$-gon is polygon with $n$ sides that are tangential for incircle and chordal for circumcircle. The connection between the radius $(R)$ of circumcirle, radius $(r)$ of incircle and distance $(d)$ of their centers represents the Fuss' relation for the certain bicentric $n$-gon. A bicentric polygon whose sides describe the inscribed circle $k$ times is called a $k$-bicentric polygon. For $k=1$ a 1-bicentric $n$-gon is simply called a bicentric $n$-gon. The relation $F_n^{(k)}left(R,r,dright)=0$ is the Fuss' relation for a $k$-bicentric $n$-gon. To date, Fuss' relations are known for bicentric $n$-gons where $n=3,4,5,6,7,8,9,10,11,12,13,14,15,16,18,20$. We present the results we obtained for the bicentric tridecagon and the pentadecagon published in the paper - M. Orlć Bachler, Z. Kaliman: Fuss' relation for bicentric tridecagon and bicentric pentadecagon, Math. Panonica, 162-170 (2024), and the new results for the bicentric septadecagon. All results were obtained using the program Wolfram Mathematica.
bicentric polygons, tridecagon, pentadecagon, septadecagon, Fuss' relation
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