Proof of Lemma 2.     It is enough to observe, according to the figure, the values of the angles of the three triangles ABC ( red color ), A'B'C' ( blue), A''B''C''( black). If  2m and 2n are the value of the indicated arcs corresponding to the two successive transformations, then those angles are: A,B,C
A'=A-2m, B'=B+m, C'=C+m
A''=A-2m+n, B"=B+m-2n, C''=C+m+n It is now easy to see that we can choose m and n such that  A''=B''=C''=60º (for that it is enough to take m = (180-B-2C)/3, n = (B - C)/3). Then the value of the angle between  B'' C'' and BC is exactly (C-B)/3. Thus  A''B''C'' has then its sides parallel to those of the Morley triangle of ABC.