Efficient method to find patterns $\frac{a}{b}=k \frac{c}{d}$ in the set

I have a list of rules, the whole set is bigger, this is just a small part

rule1=MapIndexed[Subscript[A, First@#2]->#&,{0.017840140244041125,0.03701275157916095,0.051660116184981536,0.08951663681268851,0.11158027392659164,0.11199090055895211,0.11722479640123873,0.12880796041375156,0.13998629375261973,0.14799408926444602,0.1603406353916348,0.1691630967698906,0.17603820358593392,0.17625123467649714,0.19645343424258566,0.21551707277352605,0.2181102569613515,0.23391774392335418,0.2550771120859639,0.268556465535236,0.3048461639996855,0.31092372219882125,0.3127938111713915,0.31392896827464023,0.3409781586745735,0.34596915956848695,0.3529110816686192,0.3616255919094322,0.367472697175696,0.3934402946502719,0.4047693032887368,0.4311153156718978,0.4683757162129396,0.46952099223657906,0.5072262199843859,0.5221019133548683,0.5343944162249954,0.547890105814087,0.5599232760856494,0.5668852126411741,0.5798751142315612,0.5934466012255624,0.5974131205016744,0.6095346242098095,0.6435968663448772,0.6590218413254265,0.6941771182743655,0.7420644764943874,0.7438917323215621,0.7969894568033051,0.8000537128029253,0.8374749504134635,0.8407436273863764,0.8616085956899902,0.9112275732642781,0.9275783068606626,0.9389371022951489,0.9437334963687791,0.9593199516799973,1}]; 

a, b, c, d are the four numbers in the set, and at most two are equal, satisfy the following relationship:
$$\frac{a}{b}=k \frac{c}{d}$$
$$k$$ is a rational number.
when $$k=1$$, The GroupBy based approach is fast

GroupBy[#/#2 -> (#/#2 /. rule1) & @@@ Subsets[Keys@rule1, {2}],    Round[Last@#, 10^-10.] & -> First, Equal @@ # &] // DeleteCases[True] // AbsoluteTiming 

when $$k\neq 1$$, assume that $$k = 1/3$$
Itâ€™s not easy to do with GroupBy, I thought of a way to use Gather, It’s much slower than the previous one

Gather[#/#2 -> (#/#2 /. rule1) & @@@ Subsets[Keys@rule1, {2}],    Abs[#[[2]] #2[[2]] - 1/3] < 10^-10. &] // Select[Length@# > 1 &] // AbsoluteTiming `

I want to know if $$k\neq 1$$, is there a way to use Groupby instead of Gather, or other more efficient methods?