Kosnita's theorem

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

X(54) is the Kosnita point of the triangle ABC

Let ${\displaystyle ABC}$ be an arbitrary triangle, ${\displaystyle O}$ its circumcenter and ${\displaystyle O_{a},O_{b},O_{c}}$ are the circumcenters of three triangles ${\displaystyle OBC}$, ${\displaystyle OCA}$, and ${\displaystyle OAB}$ respectively. The theorem claims that the three straight lines ${\displaystyle AO_{a}}$, ${\displaystyle BO_{b}}$, and ${\displaystyle CO_{c}}$ 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 ${\displaystyle X(54)}$ in Clark Kimberling's list.^[5] This theorem is a 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 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.