Can't you get ɛ as small as you want if you are allow arbitrary large sides? Not that it is easy but still.
So if there is no bound for the size of your triangle, this will take forever.
edit There is a simple to way to get insanely large answers which will give you ɛ as small as you want.
Check (n,n+408,577) for n as large as you want. For n large the angle tends to
180 - arccos 408/577 which gives you ɛ ~ 0.0000860...
and you can replace 408/577 by the any fraction approximating sqrt(2) / 2
So if there is no bound for the size of your triangle, this will take forever.
edit There is a simple to way to get insanely large answers which will give you ɛ as small as you want.
Check (n,n+408,577) for n as large as you want. For n large the angle tends to
180 - arccos 408/577 which gives you ɛ ~ 0.0000860...
and you can replace 408/577 by the any fraction approximating sqrt(2) / 2
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if the sides are 70 , 38 , 38 then < C ≈ 134.16°