Proof of Lemma 1(1).     In the figure QS determines the Wallace-Simson line w of P with respect to ABC and Q'S'  the W-S line  w' of P with respect to A'B'C'. Since PQSB is a cyclic quadrilateral, it is clear that the angles PSQ and PBQ are equal. In the same way the angle P'S'Q' equals PBQ. Therefore w' is obtained from w by means of a parallel translation of vector v correspondig to SS'.