Constructing equidissections for certain classes of trapezoids

We investigate equidissections of a trapezoid T(a)T(a), where the ratio of the lengths of two parallel sides is aa. (An equidissection   is a dissection into triangles of equal areas.) An integer nn is in the spectrum  S(T(a))S(T(a)) if T(a)T(a) admits an equidissection into nn triangles. Suppose aa is algebraic of degree 2 or 3, with each conjugate over Q having positive real part. We show that if nn is large enough, nn is in S(T(a))S(T(a)) iff n/(1+a)n/(1+a) is an algebraic integer. If, in addition, aa is the larger root of a monic quadratic polynomial with integer coefficients, we give a complete description of S(T(a))S(T(a)).