Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...

, Kosnita's theorem is a property of certain

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...

s associated with an arbitrary

triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...

. Let

ABC

be an arbitrary triangle,

O

its

circumcenter In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...

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 line In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are one-dimensional objects, though they may exist in two, three, or higher dimension spaces. The word ''line'' may also refer to a line segmen ...

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 __notoc__ In geometry, the isogonal conjugate of a point with respect to a triangle is constructed by reflecting the lines about the angle bisectors of respectively. These three reflected lines concur at the isogonal conjugate of . (Th ...

of the

nine-point center In geometry, the nine-point center is a triangle center, a point defined from a given triangle in a way that does not depend on the placement or scale of the triangle. It is so called because it is the center of the nine-point circle, a circle t ...

. It is

triangle center In geometry, a triangle center (or triangle centre) is a point in the plane that is in some sense a center of a triangle akin to the centers of squares and circles, that is, a point that is in the middle of the figure by some measure. For exampl ...

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.The extension from a circle to a conic having center: The creative method of new theorems
International Journal of Computer Discovered Mathematics, pp.21-32.

References

Ion Pătraşcu (2010),
A generalization of Kosnita's theorem
' (in Romanian) John Rigby (1997), ''Brief notes on some forgotten geometrical theorems.'' Mathematics and Informatics Quarterly, volume 7, pages 156-158 (as cited by Kimberling). Darij Grinberg (2003),
On the Kosnita Point and the Reflection Triangle
'' Forum Geometricorum, volume 3, pages 105–111. Clark Kimberling (2014),
Encyclopedia of Triangle Centers
'', section ''X(54) = Kosnita Point''. Accessed on 2014-10-08 Nikolaos Dergiades (2014),
Dao's Theorem on Six Circumcenters associated with a Cyclic Hexagon
'' Forum Geometricorum, volume 14, pages=243–246. . 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. . Ngo Quang Duong, International Journal of Computer Discovered Mathematics, Some problems around the Dao's theorem on six circumcenters associated with a cyclic hexagon configuration
volume 1, pages=25-39. Clark Kimberling (2014)

/ref> {{mathworld, id=KosnitaTheorem, title=Kosnita Theorem Theorems about triangles and circles