  We study a combinatorial problem that recently arose in the context of shape
optimization: among all triangles with vertices $(0,0)$, $(x,0)$, and $(0,y)$
and fixed area, which one encloses the most lattice points from
$\mathbb{Z}_{>0}^2$? Moreover, does its shape necessarily converge to the
isosceles triangle $(x=y)$ as the area becomes large? Laugesen and Liu
suggested that, in contrast to similar problems, there might not be a limiting
shape. We prove that the limiting set is indeed nontrivial and contains
infinitely many elements. We also show that there exist `bad' areas where no
triangle is particularly good at capturing lattice points and show that there
exists an infinite set of slopes $y/x$ such that any associated triangle
captures more lattice points than any other fixed triangle for infinitely many
(and arbitrarily large) areas; this set of slopes is a fractal subset of $[1/3,
3]$ and has Minkowski dimension at most $3/4$.
