Kosnita's theorem explained

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).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.^{[1] [2] [3]}

References

1. Nguyễn Minh Hà, Another Purely Synthetic Proof of Dao's Theorem on Sixcircumcenters. Journal of Advanced Research on Classical and Modern Geometries,, volume 6, pages 37–44.
2. Nguyễn Tiến Dũng, A Simple proof of Dao's Theorem on Sixcircumcenters. Journal of Advanced Research on Classical and Modern Geometries,, volume 6, pages 58–61.
3. http://www.journal-1.eu/2016-3/Nguyen-Ngoc-Giang-The-extension-pp.21-32.pdf The extension from a circle to a conic having center: The creative method of new theorems

^[4]

^[5]

^[6]

^[7]

^[8]

^[9]

^[10]

^[11]