Diophantine Steiner triples

Abstract

V članku opišemo vse trojice ▫$(R, r, d)$▫, ▫$R > r+d$▫, naravnih števil, za katere ima konfiguracija dveh krogov z radiji ▫$R$▫ in ▫$r$▫ ter razdaljo ▫$d$▫ med središči sklenjeno Steinerjevo verigo. To pomeni, da obstaja cilkično zaporedje ▫$n$▫ krogov, ki se dotikajo začetnih dveh krogov in se dotikajo sosednjih krogov v cikličnem zaporedju. Izkaže se, da je v primeru naravnih vrednosti ▫$R$▫, ▫$r$▫ in ▫$d$▫ dolžina ▫$n$▫ Steinerjeve verige lahko le 3, 4 ali 6.

Keywords

matematika;diofantske Steinerjeve trojice;Steinerjev porizem;Steinerjeva formula;diofantske enačbe;mathematics;dDiophantine Steiner triples;Steiner porism;Steiner identity;diophantine equations;

Data

Language:	English
Year of publishing:	2010
Typology:	0 - Not set
Organization:	UM FNM - Faculty of Natural Sciences and Mathematics
UDC:	514.112:511
COBISS:	15534681
ISSN:	2232-2094
Parent publication:	Preprint series
Views:	68
Downloads:	9
Average score:	0 (0 votes)
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Other data

Secondary language:	Slovenian
Secondary title:	Diofantske Steinerjeve trojice
Secondary abstract:	We describe all integer triples ▫$(R,r,d)$▫, ▫$R > r + d$▫, for which a configuration of two circles of radii ▫$R$▫ and ▫$r$▫ with the centers ▫$d$▫ apart possesses a closed Steiner chain. This means that there exists a cyclic sequence of ▫$n$▫ circles which are tangent to the starting two circles, and each of them is tangent to its two neighbors in the sequence. It appears that in the case of integer valued ▫$R$▫, ▫$r$▫ and ▫$d$▫, the only possible values for the lengths ▫$n$▫ of the Steiner chains are 3, 4 or 6.
Secondary keywords:	matematika;diofantske Steinerjeve trojice;Steinerjev porizem;Steinerjeva formula;diofantske enačbe;
URN:	URN:SI:UM:
Type (COBISS):	Not categorized
Pages:	str. 1-8
Volume:	ǂVol. ǂ48
Issue:	ǂšt. ǂ1117
Chronology:	2010
ID:	68345