Tilings of triangles

Let T be a non-equilateral triangle. We prove that the number of non-similar triangles Δ such that T can be dissected into triangles similar to Δ is at most 6. On the other hand, for infinitely many triangles T there are six non-similar triangles Δ such that T can be dissected into congruent triangles similar to Δ. For the equilateral triangle there are infinitely many such Δ. We also investigate the number of pieces in the dissections of the equilateral triangle into congruent triangles.