  Let $\cal T$ be a tiling of the plane with equilateral triangles no two of
which share a side. We prove that if the side lengths of the triangles are
bounded from below by a positive constant, then $\cal T$ is periodic and it
consists of translates of only at most three different triangles. As a
corollary, we prove a theorem of Scherer and answer a question of Nandakumar.
The same result has been obtained independently by Richter and Wirth.
