  In this paper, we show that the maximum number of points in $d\geq3$
dimensions determining exactly 2 distinct triangles is $2d$. We further show
that this maximum is uniquely achieved by the vertices of the $d$-orthoplex. We
build upon the work of Hirasaka and Shinohara who determined that the
$d$-orthoplex is such an optimal configuration, but did not prove its
uniqueness. Further, we present a more elementary argument for its optimality.
