Given two (Euclidean or hyperbolic) triangles $ T = ABC $ and $ T' = A'B'C' $,
the natural map is the one that sends
    $ A' \mapsto A $,
    $ B' \mapsto B $,
    $ C' \mapsto C $
    and maps affinely each side of $T'$
onto the corresponding side of $T$.
    We say that the triangle $ T' $ dominates
    the triangle $ T $
    if the natural map is a short map (Lipschitz with constant $1$) with respect to the distance in the (Euclidean or hyperbolic) plane.

My question is, given a triangle $T$ with side lengths $(a, b,c)$,
is it true that the triangle $T'$ with side lengths $(a+\epsilon, b+\epsilon, c+\epsilon)$ dominates $T$ for all $\epsilon>0$ small enough?

I can prove this statement for Euclidean triangles by some calculations involving the law of cosines, but I couldn't manage to do
the same in the hyperbolic plane.