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.



Short map between hyperbolic triangles
No, this is not true. Here is an indirect argument (if I made no mistake).


If the statement would be true for $\epsilon > 0$ small enough, then it would be true for all $\epsilon > 0$ (since the maps commute, the set of "good" epsilons for a given $(a,b,c)$ is open; this set is also closed because the Lipschitz condition is closed). We will construct two triangles $(a,b,c)$ and $(a+M, b+M, c+M)$ for which the latter does not dominate the former.


Take $a=b=1$ and $c=2$ (or slightly smaller if you need). Take points $A\_1$ and $B\_1$ on the sides $BC$ and $AC$ at distance $\frac13$ from $C$. The distance between $A\_1$ and $B\_1$ is $\frac23$. Now consider a triangle $(x, x, x+1)$ for a very large $x$. I claim that the distance between the corresponding points $A'\_1$ and $B'\_1$ tends to $0$ as $x$ tends to $\infty$.


By the sine law in a right triangle we have
$$\frac{\sinh \frac{A'\_1B'\_1}{2}}{\sinh\frac{x}{3}} = \sin\frac{\gamma'}{2} = \frac{\sinh\frac{x+1}2}{\sinh x}$$
which implies that $\sinh\frac{A'\_1B'\_1}2$ goes down as $e^{-\frac{x}6}$.


This means that somewhere inbetween there is an isosceles triangle for which the map is not $1$-Lipschitz even for small $\epsilon$...


