Ratios which are the same with the same ratio are also the same with one another.
Proof
If a:b=c:d and c:d=e:f, then for any equimultiples the same
inequality test holds for (a,b) as for (c,d), and that same test
holds for (c,d) as for (e,f); hence by transitivity of the
inequality test, the test holds for (a,b) versus (e,f).