Given three points in the plane, interest is in the locus of all points for which the sum of the distances to the given points is a prescribed constant. These curves turn out to be sixth degree polynominals in x and y , and thus are complicated. However, it turns out that often there is a point, within the triangle formed by the three given points, for which the constant is a minimum. This point is determined for isosceles triangles, the general case is discussed, although all details are not revealed. Student problems, such as supplying the missing details mentioned and further generalizations, are suggested. (Contains 10 figures.)