Sadi Abu-Saymeh and Mowaffaq Hajja, Coincidence of centers for
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).