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 $ABC$ 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. 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 $X(54)$ in Clark Kimberling's list. This theorem is a special case of Dao's theorem on six circumcenters associated with a cyclic hexagon in.