Points on Hyperbolas at Rational Distance

Richard Guy asked for the largest set of points which can be placed in the plane so that their pairwise distances are rational numbers. In this article, we consider such a set of rational points restricted to a given hyperbola. To be precise, for rational numbers $a$, $b$, $c$, and $d$ such that the quantity $D = \bigl(a \, d - b \, c \bigr) / \bigl(2 \, a^2 \bigr)$ is defined and nonzero, we consider rational distance sets on the conic section $a \, x \, y + b \, x + c \, y + d = 0$. We show that, if the elliptic curve $Y^2 = X^3 - D^2 \, X$ has infinitely many rational points, then there are infinitely many sets consisting of four rational points on the hyperbola such that their pairwise distances are rational numbers. We also show that any rational distance set of three such points can always be extended to a rational distance set of four such points.