The following beautiful result is named after the Romanian mathematician Cezar Coşniţă (1910-1962). I first saw this theorem in the September 1995 issue of the Mathematics and Informatics Quarterly, in the column on Forgotten Theorems.
Kosnita's Theorem
The lines joining the vertices A, B, and C of a given triangle ABC with the respective circumcentres D, E and F of the triangles ABO, BCO, and CAO (where O is the circumcentre of ABC) are concurrent.
Challenge
Can you prove the result? Can you prove it in more than one way?
Kosnita's Theorem
Proof Hint
1) The result can be proved in several different ways, but probably the easiest way to prove it is to use Jacobi's theorem.
2) Try to use Jacobi's theorem at the link above to prove Kosnita's theorem.
3) Check: Compare your solution in 2 above with the proof of Kosnita given in De Villiers (1996) - proof is towards the end of the paper.
A Dual to Kosnita
Due to the often observed duality between circumcentres and incentres as explored fairly extensively in De Villiers (1994), it seems natural to expect a dual to Kosnita's theorem involving incentres.
4) Can you formulate a dual to Kosnita's theorem involving incentres? Can you prove it?
5) Check: Compare your formulated dual and proof of Kosnita with that given in De Villiers (1996). Or see Dual to Kosnita (De Villiers points of a triangle).
Further Generalization
It turns out that Kosnita's theorem is a special case of a more general theorem about a cyclic hexagon discovered by Dao Thanh Oai - see for example, Ha (2017).
References
De Villiers, M. (1994, 1996, 2009). Some Adventures in Euclidean Geometry. (Free to download). Lulu Publishers: Dynamic Mathematics Learning.
De Villiers, M. (1996). A Dual to Kosnita's theorem. Mathematics & Informatics Quarterly, 6(3), 169-171.
Ha, N.M. (2017). Another Purely Synthetic Proof of Dao's Theorem on Six Circumcenters. Global Journal of Advanced Research on Classical and Modern Geometries, Vol.6, Issue 1, pp.37-44.
Related Links
Fermat-Torricelli Point Generalization (aka Jacobi's theorem) plus Further Generalizations
Weighted Airport Problem
Dual to Kosnita (De Villiers points of a triangle)
Another concurrency related to the Fermat point of a triangle plus related results
Water Supply II: Three Towns (Rethinking Proof activity)
The Center of Gravity of a Triangle (Rethinking Proof activity)
Concurrency Conjecture (Rethinking Proof activity)
Triangle Altitudes (Rethinking Proof activity)
The Fermat-Torricelli Point (Rethinking Proof activity)
Airport Problem (Rethinking Proof activity)
Napoleon (Rethinking Proof activity)
Miquel (Rethinking Proof activity)
Experimentally Finding the Medians and Centroid of a Triangle
Experimentally Finding the Centroid of a Triangle with Different Weights at the Vertices (Ceva's theorem)
Point Mass Centroid (centre of gravity or balancing point) of Quadrilateral
Triangle Centroids of a Hexagon form a Parallelo-Hexagon: A generalization of Varignon's Theorem
Nine-point centre & Maltitudes of Cyclic Quadrilateral
A side trisection triangle concurrency
A 1999 British Mathematics Olympiad Problem and its dual
Concurrency and Euler line locus result
Bride's Chair Concurrency
Haag Hexagon - Extra Properties
Concurrency, collinearity and other properties of a particular hexagon
Power Lines of a Triangle
Carnot's Perpendicularity Theorem & Some Generalizations
Generalizing the concepts of perpendicular bisectors, angle bisectors, medians and altitudes of a triangle to 3D
Anghel's Hexagon Concurrency theorem
Some Circle Concurrency Theorems
Three Overlapping Circles (Haruki's Theorem)
Parallel-Hexagon Concurrency Theorem
Toshio Seimiya Theorem: A Hexagon Concurrency result
Van Aubel's Theorem and some Generalizations (See concurrency in Similar Rectangles on sides)
The quasi-circumcentre and quasi-incentre of a quadrilateral
External Links
Kosnita Theorem (Wolfram MathWorld)
Kosnita's theorem (Wikipedia)
More on The Kosnita Point (Art of Problem Posing)
Concurrent lines (Wikipedia)
SA Mathematics Olympiad Questions and worked solutions for past South African Mathematics Olympiad papers can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)
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Michael de Villiers, created with WebSketchpad, 2 Oct 2025; updated 3 Oct 2025.