Consider two (Euclidean) triangles $T$ and $T'$.
Let's say that $T$ *majorizes* $T'$ if there exists a 1-Lipschitz map that sends
vertices to vertices and sides to sides (for some labeling of the vertices).

My question is, what are necessary and sufficient conditions for $T$ to majorize $T'$ ?

I know a sufficient condition.
Let's say that the lengths $(l_1, l_2, l_3)$ of $T$ and $(l_1', l_2', l_3')$
of $T'$ satisfy the *strong triangle inequalities* if
$l_i + l_j - l_k \ge l_i' + l_j' - l_k'$ for all pairwise distinct $i,j,k$.
Then if $T$ and $T'$ satisfy the strong triangle inequalities, then $T$ majorizes $T'$. But this condition is not necessary (see the answer on math.se).