Classification for the (2α,2β,α+β) triangle

Prove that, whenever a triangle with angles (2α,2β,α+β) can be tiled by congruent copies of the triangle (a,b,c), the number of tiles has the form (a+2b)(b+2a)m^2, and show that any counterexample requires a≡b (mod 3).

Background

The paper constructs infinite families of tilings of triangles with angles (2α,2β,α+β). Under the condition a≠b modulo 3, it proves the asserted square-form restriction on the number of tiles, and conjectures the same form in general, identifying the congruence case a≡b modulo 3 as the only possible source of counterexamples.