An equisum set in the plane is a finite set X$_n$ of distinct points $\lbrack$x$_1, x_2$, ..., $x_n\rbrack$ such that the sum of the distances from x$_i$ to all of the other points in X is the same for all i = 1, 2, ..., n. The authors show that an equisum set in which all the interpoint distances are different must have at least 5 elements. They also investigate the possible numbers of different interpoint distances in an equisum set with 5 elements.