  We consider three types of isogonal conjugacy of two points with respect to a
given triangle and characterize any of these types by a geometric equality. As
an application to the Fermat problem with positive weights, we prove that in
the general case the given weights determine uniquely a point X and the
solution to the Fermat problem is the point Y, which is isogonally conjugate of
type I to the point X. We obtain a similar characterization of the solution to
the Fermat problem in the case of mixed weights as well.
