Distinct points $P,Q,R,S$ lie on the circle $x^2+y^2=169$ and have integer coordinates. The distances $PQ$ and $RS$ are irrational. What is the greatest possible value of $\frac{PQ}{RS}$?

1. The lattice points on this circle are $(\pm13,0),(0,\pm13),(\pm5,\pm12),(\pm12,\pm5)$.
2. Checking all pairwise distances, the shortest irrational one is $\sqrt{26}$ (e.g. between $(-13,0)$ and $(-12,-5)$), and the longest is $5\sqrt{26}$ (e.g. between $(0,13)$ and $(5,-12)$).
3. The ratio is $\boxed5$.