Kosnita's theorem

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

X(54) is Kosnita point of the triangle ABC

Let be an arbitrary triangle, its circumcenter and are the circumcenters of three triangles , , and respectively. The theorem claims that the three straight lines , , and are concurrent.[1] This result was established by the Romanian mathematician Cezar Coşniţă (1910-1962).[2]

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.[3][4] It is triangle center in Clark Kimberling's list.[5] This theorem is special case of Dao's theorem on six circumcenters associated with a cyclic hexagon in.[6][7][8][9][10][11][12]

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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