We define a triangle design as a partition of the set of lines of a projective space into triangles, where a triangle consists of three pairwise intersecting lines with no common point. A triangle design is balanced if all points are involved in the same number of triangles. We construct balanced triangle designs in PG$(n-1,2)$ for all admissible $n$ (congruent to $1$ modulo $6$) and an infinite class of balanced block-divisible triangle designs. We also prove that the existence of a triangle design in PG$(n-1,2)$ invariant under the action of the Singer cycle group is equivalent to the existence of a partition of $Z_{2^n-1}\backslash\{0\}$ into special $18$-subsets and find such partitions for $n=7$, $13$, $19$. Keywords: Subspace design, graph decomposition, triangle design, Heffter's difference problem.