Proof  of Lemma 1(3). According to one of the classical proofs of Morley's theorem, the one by Naraniengar (see p.47 in Coxeter-Greitzer, Geometry Revisited, MAA, Washington), the side ZY of the Morley triangle close to A is at an angle (C-B)/3 with BC. But one easily sees that for the two triangles  ABC and AB'C' one has (C'-B')/3=(C-B)/3 and thus the two Morley triangles of  ABC and A'B'C' are as in the statement of the lemma.