Research Article
BibTex RIS Cite
Year 2021, , 226 - 230, 15.04.2021
https://doi.org/10.36890/iejg.748127

Abstract

References

  • [1] Bakers, H. F.:The principles of Geometry. Vol I Foundations, Cambridge at the University Press, (1929).
  • [2] Bataille, M. : Correction of the problem 3945 Proposed by J. Chris Fisher. Crux Mathematicorum. 41/5, May Solutions 221 (2015).
  • [3] Cohl, T.: A purely synthetic proof of Dao’s theorem on six circumcenters associated with a cyclic hexagon. Forum Geom. 14, 261-264 (2014).
  • [4] Dao, O.T.: Problem 3845. Crux Mathematicorum. 39, Issue May (2013).
  • [5] Dergiades, N.: Dao’s Theorem on Six Circumcenters associated with a Cyclic Hexagon. Forum Geom. 14, 243-246 2014).
  • [6] Dung, N.T.: A simple proof of Dao’s theorem on sic circles . Global Journal of Advanced Research on Classical and Modern Geometries. 6/1, 58–61 (2017).
  • [7] Glaeser, G, Stachel, H. and Odehnal, B.: The Universe of Conics. SpringerSpektrum, Springer-Verlag Berlin Heidelberg, (2016).
  • [8] Gévay, G.: An extension of Miquel’s six-circles theorem. Forum Geometricorum. 18 , 115–118 (2018).
  • [9] Gévay, G.: Resolvable configurations. Discrete Applied Mathematics. 266, 319-330 (2019).
  • [10] Ngo, O. D.: Some problems around the Dao’s theorem on six circumcenters associated with a cyclic hexagon configuration. International Journal of Computer Discovered Mathematics. 1/2, 40–47 (2016).
  • [11] Miquel, A.: Theoremes sur les intersections des cercles et des spheres. Journal de mathematiques pures et appliquees Ire serie. 3, 517–522 (1838).

A Note on Centres in a Chain of Circles

Year 2021, , 226 - 230, 15.04.2021
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.

References

  • [1] Bakers, H. F.:The principles of Geometry. Vol I Foundations, Cambridge at the University Press, (1929).
  • [2] Bataille, M. : Correction of the problem 3945 Proposed by J. Chris Fisher. Crux Mathematicorum. 41/5, May Solutions 221 (2015).
  • [3] Cohl, T.: A purely synthetic proof of Dao’s theorem on six circumcenters associated with a cyclic hexagon. Forum Geom. 14, 261-264 (2014).
  • [4] Dao, O.T.: Problem 3845. Crux Mathematicorum. 39, Issue May (2013).
  • [5] Dergiades, N.: Dao’s Theorem on Six Circumcenters associated with a Cyclic Hexagon. Forum Geom. 14, 243-246 2014).
  • [6] Dung, N.T.: A simple proof of Dao’s theorem on sic circles . Global Journal of Advanced Research on Classical and Modern Geometries. 6/1, 58–61 (2017).
  • [7] Glaeser, G, Stachel, H. and Odehnal, B.: The Universe of Conics. SpringerSpektrum, Springer-Verlag Berlin Heidelberg, (2016).
  • [8] Gévay, G.: An extension of Miquel’s six-circles theorem. Forum Geometricorum. 18 , 115–118 (2018).
  • [9] Gévay, G.: Resolvable configurations. Discrete Applied Mathematics. 266, 319-330 (2019).
  • [10] Ngo, O. D.: Some problems around the Dao’s theorem on six circumcenters associated with a cyclic hexagon configuration. International Journal of Computer Discovered Mathematics. 1/2, 40–47 (2016).
  • [11] Miquel, A.: Theoremes sur les intersections des cercles et des spheres. Journal de mathematiques pures et appliquees Ire serie. 3, 517–522 (1838).
There are 11 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Akos G.horváth

Publication Date April 15, 2021
Acceptance Date February 26, 2021
Published in Issue Year 2021

Cite

APA G.horváth, A. (2021). A Note on Centres in a Chain of Circles. International Electronic Journal of Geometry, 14(1), 226-230. https://doi.org/10.36890/iejg.748127
AMA G.horváth A. A Note on Centres in a Chain of Circles. Int. Electron. J. Geom. April 2021;14(1):226-230. doi:10.36890/iejg.748127
Chicago G.horváth, Akos. “A Note on Centres in a Chain of Circles”. International Electronic Journal of Geometry 14, no. 1 (April 2021): 226-30. https://doi.org/10.36890/iejg.748127.
EndNote G.horváth A (April 1, 2021) A Note on Centres in a Chain of Circles. International Electronic Journal of Geometry 14 1 226–230.
IEEE A. G.horváth, “A Note on Centres in a Chain of Circles”, Int. Electron. J. Geom., vol. 14, no. 1, pp. 226–230, 2021, doi: 10.36890/iejg.748127.
ISNAD G.horváth, Akos. “A Note on Centres in a Chain of Circles”. International Electronic Journal of Geometry 14/1 (April 2021), 226-230. https://doi.org/10.36890/iejg.748127.
JAMA G.horváth A. A Note on Centres in a Chain of Circles. Int. Electron. J. Geom. 2021;14:226–230.
MLA G.horváth, Akos. “A Note on Centres in a Chain of Circles”. International Electronic Journal of Geometry, vol. 14, no. 1, 2021, pp. 226-30, doi:10.36890/iejg.748127.
Vancouver G.horváth A. A Note on Centres in a Chain of Circles. Int. Electron. J. Geom. 2021;14(1):226-30.