The Rational Distance Problem for Isosceles Triangles with one rational side

For a triangle $\Delta$, let (P) denote the problem of the existence of points in the plane of $\Delta$, that are at rational distance to the 3 vertices of $\Delta$. Answer to (P) is known to be positive in the following situation: $\Delta$ has one rational side and the square of all sides are rational. Further, the set of solution-points is dense in the plane of $\Delta$. See [3] The reader can convince himself that the rationality of one side is a reasonable minimum condition to set out, otherwise problem (P) would stay somewhat hazy and scattered. Now, even with the assumption of one rational side, problem (P) stays hard. In this note, we restrict our attention to isosceles triangles, and provide a \textit{complete description} of such triangles for which (P) has a positive answer.