In this paper the Weber problem withnexisting facilities in the Euclidean plane is considered. The solution pointX*(x*,y*)can be distinct from the existing facilities (isolated point) or coinciding with one of them (coinciding point), so that there aren+ 1 different types of possible solutions. In the paper the space of the weights is partitioned inton +1 regionsRjcorresponding to them, and a graphic construction of theRjis given for the casesn= 3 andn =4.