Lambert-saccheri Quadrangles

Starting from a general absolute plane (E, Q,-α) in the sense of [2] p. 96¹ H. Karzel and M.Marchi in [1] introduced the notion of a Lambert-Saccheri quadrangle (LS quadrangle): Any quadruple (a, b, c, d) of points of E , no three collinear, is a LS quadrangle, if Denoting the foot of a on the line d with a' := the LS quadrangle (a, b, c, d) is called rectangle, hyperbolic or elliptic quadrangle if a' = d,a' \in ] c, d[or a' \notiņ]c, d [\bigcup\d\ respectively. Let LS be the set of all LS quadrangles and LS _r , LS _h or LS _e the subset of all rectangles, hyperbolic or elliptic LS quadrangles respectively. In [1] it was claimed that either LS = LS _r or LS = LS _h or LS = LS _e . But in [1], (2.3) the proof of one special case is missing, namely that LS quadrangles (a, b, c, d) and (c, b, a, d) are of the same LS type. This note will supplement the missing part. We recall that LS_r ≠ 0 implies LS = LS _r (cf. [2] (21.3)) and therefore we will only consider the case LS_r ≠ 0, i.e. \overlinec,d} \not\perp \overlined, a .