TILINGS OF PARALLELOGRAMS WITH SIMILAR TRIANGLES

We say that a triangle Delta tiles the polygon P if P can be decomposed into finitely many non-overlapping triangles similar to Delta. Let P be a parallelogram with angles delta and pi - delta (0 < delta <= pi/2) and let Delta be a triangle with angles alpha, beta, gamma (alpha <= beta <= gamma). We prove that if Delta tiles P then either delta is an element of {alpha, beta, gamma, pi - gamma, pi - 2 gamma} or dimL(P) = dimL(Delta). We also prove that for every parallelogram P, and for every integer n (where n >= 2, n not equal 3) there is a triangle Delta so that n similar copies of Delta tile P.