A few years ago, I saw this problem solved in small russian book author Y.A.Uspenki. The solution is based in the minimum potential energy for the three masses. I am more than sure Alex will know about it and he can explain it in better words than me.

The problem in the book is referred where to locate a facility building, so the total distance from three villages were minimal, considering different number of inhabitants for each village and it is solved using weights proportional to each population.

The Fermat point is obviously a particular case of it, since all the population are the same.