  Given $\eta=\begin{pmatrix}
  a&b\\c&d
  \end{pmatrix}\in \text{GL}_2(\mathbb{Q})$, we consider the number of rational
points on the genus one curve
  \[H_\eta:y^2=(a(1-x^2)+b(2x))^2+(c(1-x^2)+d(2x))^2.\]
  We prove that the set of $\eta$ for which $H_\eta(\mathbb{Q})\neq\emptyset$
has density zero, and that if a rational point $(x_0,y_0)\in
H_\eta(\mathbb{Q})$ exists, then $H_\eta(\mathbb{Q})$ is infinite unless a
certain explicit polynomial in $a,b,c,d,x_0,y_0$ vanishes.
  Curves of the form $H_\eta$ naturally occur in the study of configurations of
points in $\mathbb{R}^n$ with rational distances between them. As one example
demonstrating this framework, we prove that if a line through the origin in
$\mathbb{R}^2$ passes through a rational point on the unit circle, then it
contains a dense set of points $P$ such that the distances from $P$ to each of
the three points $(0,0)$, $(0,1)$, and $(1,1)$ are all rational. We also prove
some results regarding whether a rational number can be expressed as a sum or
product of slopes of rational right triangles.
