Kosnita's theorem



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. 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.