Kosnita's theorem

X(54) is Kosnita point of the triangle ABC

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

Let $ABC$ be an arbitrary triangle, $O$ its circumcenter and $O_a,O_b,O_c$ are the circumcenters of three triangles $OBC$, $OCA$, and $OAB$ respectively. The theorem claims that the three straight lines $AO_a$, $BO_b$, and $CO_c$ are concurrent.^[1]

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.^[2][3] It is triangle center $X(54)$ in Clark Kimberling's list.^[4] This theorem is special case of Dao's theorem on six circumcenters associated with a cyclic hexagon in.^[5][6]

References

1. Weisstein, Eric W., "Kosnita Theorem", MathWorld.
2. Darij Grinberg (2003), On the Kosnita Point and the Reflection Triangle. Forum Geometricorum, volume 3, pages 105–111. ISSN 1534-1178
3. John Rigby (1997), Brief notes on some forgotten geometrical theorems. Mathematics and Informatics Quarterly, volume 7, pages 156-158 (as cited by Kimberling).
4. Clark Kimberling (2014), Encyclopedia of Triangle Centers, section X(54) = Kosnita Point. Accessed on 2014-10-08
5. Nikolaos Dergiades (2014), Dao’s Theorem on Six Circumcenters associated with a Cyclic Hexagon. Forum Geometricorum, volume 14, pages=243–246. ISSN 1534-1178.
6. 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.