Given a triangle (or, in general, a simplex) $T$ and a vector $\vec{s}$, I'd like to compute the quantity $$ \max\{|x-y|: x,y\in T, x-y = \alpha \vec{s}, \alpha\in\mathbb{R}\} $$ i.e., the maximum length of a scaled vector that fits into the triangle.
I was about to start fiddling with the angles, but I thought there might be a smarter way to get the job done.