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.