How can I eliminate a parameter by substitution and simplification?

I have this equation

k = (a b c t)/(Sqrt[(f + t)^2])


and I want to set

p = t/(t + f)


In which way I can eliminate t/(t+f)?

I tried this:

FullSimplify[K, TransformationFunctions -> {(# /. t/(t + f) -> p &), Automatic}]


But it did not work.

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Eliminate[{
k == (a b c t)/(Sqrt[(f + t)^2]),
p == t/(t + f)
},
{t, f}]

a^2 b^2 c^2 p^2 == k^2

k = (a b c t)/(Sqrt[(f + t)^2]);


Assuming that {t > 0, p > 0}

Simplify[k /. t + f -> t/p, {t > 0, p > 0}]

(*  a b c p  *)


Or,

Simplify[k /. f -> t (1 - p)/p, {t > 0, p > 0}]

(*  a b c p  *)


Or, assuming {0 <= p < 1, f > 0}

Simplify[k /. t -> f p/(1 - p), {0 <= p < 1, f > 0}]

(*  a b c p  *)