Kosnita's theorem

Kosnita point of an arbitrary triangle.
  Triangle △ABC
  Circumcircle (centered at circumcenter O)
  Johnson circles (concur at O)
  AOa, BOb, COc (concur at Kosnita point)

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 Oa, Ob, Oc are the circumcenters of three triangles △OBC, △OCA, and △OAB respectively. The theorem asserts that the three straight lines AOa, BOb, and COc 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 X(54) in Clark Kimberling's list.[5]

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" (PDF). Forum Geometricorum. 3: 105–111.
  4. ↑ John Rigby (1997). "Brief notes on some forgotten geometrical theorems". Mathematics and Informatics Quarterly. 7: 156–158. (as cited by Kimberling).
  5. ↑ Clark Kimberling (2014), Encyclopedia of Triangle Centers, section X(54) = Kosnita Point. Accessed on 2014-10-08. Archived 2012-04-19 at the Wayback Machine


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