FullSimplify fails with simple assumptions

I have a list of relations of the form

x[1] x[2] + y[1] y[2] - z[1] z[2] == 0 

and I want Mathematica to simplify certain expressions using these relations. I have been using FullSimplify with assumptions, and I have found that it doesn’t work occasionally. For example:

FullSimplify[(a b - e f)/c, Assumptions -> {a b + c d - e f == 0}] 

works just fine, and gives the correct answer -d, but for some reason

FullSimplify[(c d - e f)/a, Assumptions -> {a b + c d - e f == 0}] 

returns (c d - e f)/a. Any idea why this happens and how to make sure that Mathematica recognizes the second example as -b?