A class of major centers of triangles (Q1390822)

The author shows how techniques in functional equations can be used to solve some geometrical problems involving centers of triangles. Precisely, if one considers the angles \(A, B, C \) of a triangle as real variables then a triangle center \(Y\) is said to be a major center if there exists a function \(f\) such that \(Y\) is given in trilinear coordinates by \( Y= f(A) : f(B): f(C)\). Then it is shown that if \(f\) satisfies \( f ( \pi - A) = - f(A)\) then it is possible to prove that certain lines \( A Y_A \), \( BY_B\), \( C Y_C\) obtained by evaluating \(Y\) in three different triangles are concurrent. Trilinear coordinates for the point of concurrence are given. This and other previous papers of the author show how very precise geometrical problems involving centers of triangles can be treated properly with today's mathematical tools.