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Kosnita's theorem

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Concurrency of lines connecting to certain circles associated with an arbitrary triangle
X(54) is the Kosnita point of the triangle ABC

In Euclidean geometry, Kosnita's theorem is a property of certain circles associated with an arbitrary triangle.

Let A B C {\displaystyle ABC} be an arbitrary triangle, O {\displaystyle O} its circumcenter and O a , O b , O c {\displaystyle O_{a},O_{b},O_{c}} are the circumcenters of three triangles O B C {\displaystyle OBC} , O C A {\displaystyle OCA} , and O A B {\displaystyle OAB} respectively. The theorem claims that the three straight lines A O a {\displaystyle AO_{a}} , B O b {\displaystyle BO_{b}} , and C O c {\displaystyle CO_{c}} are concurrent. This result was established by the Romanian mathematician Cezar Coşniţă (1910-1962).

Their point of concurrence is known as the triangle's Kosnita point (named by Rigby in 1997). It is the isogonal conjugate of the nine-point center. It is triangle center X ( 54 ) {\displaystyle X(54)} in Clark Kimberling's list. This theorem is a special case of Dao's theorem on six circumcenters associated with a cyclic hexagon in.

References

  1. Weisstein, Eric W. "Kosnita Theorem". MathWorld.
  2. Ion Pătraşcu (2010), A generalization of Kosnita's theorem (in Romanian)
  3. Darij Grinberg (2003), On the Kosnita Point and the Reflection Triangle. Forum Geometricorum, volume 3, pages 105–111. ISSN 1534-1178
  4. John Rigby (1997), Brief notes on some forgotten geometrical theorems. Mathematics and Informatics Quarterly, volume 7, pages 156-158 (as cited by Kimberling).
  5. Clark Kimberling (2014), Encyclopedia of Triangle Centers Archived 2012-04-19 at the Wayback Machine, section X(54) = Kosnita Point. Accessed on 2014-10-08
  6. Nikolaos Dergiades (2014), Dao's Theorem on Six Circumcenters associated with a Cyclic Hexagon. Forum Geometricorum, volume 14, pages=243–246. ISSN 1534-1178.
  7. Telv Cohl (2014), A purely synthetic proof of Dao's theorem on six circumcenters associated with a cyclic hexagon. Forum Geometricorum, volume 14, pages 261–264. ISSN 1534-1178.
  8. Ngo Quang Duong, International Journal of Computer Discovered Mathematics, Some problems around the Dao's theorem on six circumcenters associated with a cyclic hexagon configuration, volume 1, pages=25-39. ISSN 2367-7775
  9. Clark Kimberling (2014), X(3649) = KS(INTOUCH TRIANGLE)
  10. Nguyễn Minh Hà, Another Purely Synthetic Proof of Dao's Theorem on Sixcircumcenters. Journal of Advanced Research on Classical and Modern Geometries, ISSN 2284-5569, volume 6, pages 37–44. MR....
  11. Nguyễn Tiến Dũng, A Simple proof of Dao's Theorem on Sixcircumcenters. Journal of Advanced Research on Classical and Modern Geometries, ISSN 2284-5569, volume 6, pages 58–61. MR....
  12. The extension from a circle to a conic having center: The creative method of new theorems, International Journal of Computer Discovered Mathematics, pp.21-32.


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