#### Rad HAZU, Matematičke znanosti, Vol. 18 (2014), 145-170.

### AN IMPROVED METHOD FOR ESTABLISHING FUSS' RELATIONS FOR BICENTRIC *n*-GONS WHERE *n* ≥ 4
IS AN EVEN INTEGER

### Mirko Radić

University of Rijeka, Department of Mathematics, Radmile Matejčić 2, 51000 Rijeka, Croatia

*e-mail:* `mradic@ffri.hr`

**Abstract.** In [7] we have given one relatively simple and practical
method for establishing Fuss’ relations for bicentric *n*-gons where *n* ≥ 3
is an odd integer. In the present article we give one relatively simple and
practical method for establishing Fuss’ relations for bicentric *n*-gon where
*n* ≥ 4 is an even integer. In [7] the rotation numbers for bicentric *n*-gons
have the key role, while in the present article tangent lengths of bicentric
*n*-gons have the key role. So in the present article is described an algorithm
to obtain Fuss’ relation for bicentric *n*-gons where *n* ≥ 4 is an even integer.
Several yet unknown Fuss’ relations are established.

**2010 Mathematics Subject Classification.**
51E12.

**Key words and phrases.** Bicentric polygon, Fuss’ relations.

**Full text (PDF)** (free access)

**References:**

- A. Cayley,
*Developments on the porism of the in-and-circum-scribed polygon*,
Philosophical Magazine **VII** (1854) 339-345.

- N. Fuss,
*De quadrilateris quibus circulum tam inscribere quam cicumscribere licet*,
NAAPS 1792 (Nova acta), t. **X** (1797), 103-125.

- N. Fuss,
*De poligonis simmetrice irregularibus calculo simul inscriptis et circumscriptis*,
NAAPS 1792 (Nova acta), t. **XIII** (1802), 168-189.

- B. Mirman,
*Short cycles of Poncelet's conics*,
Linear Algebra Appl. **432** (2010), 2543-2564.

MathSciNet
CrossRef

- J. V. Poncelet,
*Traité des propriétés projectives des figures*,
Paris 1865 (first ed. in 1822).

- M. Radić and Z.~Kaliman,
*About one relation concerning two circles where one is inside of the other*,
Math. Maced. **3** (2005), 45-50.

MathSciNet

- M. Radić,
*An improved method for establishing Fuss' relations for bicentric polygons*,
C. R. Math. Acad. Sci. Paris **348** (2010), 415-417.

MathSciNet
CrossRef

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