Integer parabola points have pairwise distinct distances

For $b < a$ and $d < c$ in $\mathbb{N}$, if $(a-b)^2(1+(a+b)^2) = (c-d)^2(1+(c+d)^2)$ then $a = c$ and $b = d$. Hence distinct nonnegative integer points $(t,t^2)$ have pairwise distinct distances: an $n$-point set of this form is in general position and determines exactly $\binom{n}{2}$ distances. Supporting construction fact for exact distinct-distance bounds; proves no lower bound by itself.