Sadi Abu-Saymeh and Mowaffaq Hajja, Coincidence of centers for scalene trianngles,
Forum Geometricorum, 7 (2007) 137--146.

Abstract: A center function  is a function Z that assigns to every triangle T in a Euclidean plane E a point Z(T) in E in a manner that is symmetric and that respects isometries and dilations. A family F of center functions is said to be  complete if for every scalene triangle ABC and every point P in its plane, there is Z in F such that Z(ABC) = P. It is said to be separating if no two center  functions in F coincide for any scalene triangle. In this note, we give  simple examples of complete separating families of  continuous triangle center functions. Regarding the impression that no two different center functions can coincide on a scalene triangle, we show that for every center function Z and every scalene triangle T, there is another center function Z', of a simple type, such that Z(T) =Z'(T).

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