  When applied to statistical systems showing an arctic curve phenomenon, the
tangent method assumes that a modification of the most external path does not
affect the arctic curve. We strengthen this statement and also make it more
concrete by observing a factorization property: if $Z^{}_{n+k}$ denotes a
refined partition function of a system of $n+k$ non-crossing paths, with the
endpoints of the $k$ most external paths possibly displaced, then at dominant
order in $n$, it factorizes as $Z^{}_{n+k} \simeq Z^{}_{n} Z_k^{\rm out}$ where
$Z_k^{\rm out}$ is the contribution of the $k$ most external paths. Moreover if
the shape of the arctic curve is known, we find that the asymptotic value of
$Z_k^{\rm out}$ is fully computable in terms of the large deviation function
$L$ introduced in \cite{DGR19} (also called Lagrangean function). We present
detailed verifications of the factorization in the Aztec diamond and for
alternating sign matrices by using exact lattice results. Reversing the
argument, we reformulate the tangent method in a way that no longer requires an
extension of the domain, and which reveals the hidden role of the $L$ function.
As a by-product, the factorization property provides an efficient way to
conjecture the asymptotics of multirefined partition functions.
