A Note on Centres in a Chain of Circles

Year 2021, , 226 - 230, 15.04.2021

Akos G.horváth

https://doi.org/10.36890/iejg.748127

Abstract

In this note, we study a chain of circles whose pairwise intersection points, taken in a certain order, also lie on two circles. We give a short elementary proof of the following fact. There exists a conic which touches each line connecting the centres of adjacent circles of such chain. In the case of six circles of the chain, this means that the centres of these circles form a Brianchon hexagon. We consider all cases of the possible radically distinct positions of the original chain of circles. In the case when the touching conic is unique, we find out its type.

Keywords

Chain of circles, Dao's theorem, Miquel's theorem, Brianchon hexagon, inscribed conic

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