6.32Every projective collineation of period 2 is a harmonic homology.
Given a projective collineation of period 2, suppose it interchanges a pair of distinct points PP’ and also another pair QQ’. By 6.13, it is the only projective collineation that transforms the quadrangle PP’QQ’ into P’PQ’Q. The invariant lines PP’ and QQ’ meet in an invariant point O. Since the collineation interchanges the pair of lines PQ, P’Q’, and likewise the pair PQ’, P’Q, the two points M=PQ.P’Q’ and N=PQ’.P’Q are invariant. Moreover, the two invariant lines PP’ and MN meet in a third invariant point L on MN. By axiom 2.18, every point on MN is invariant. Thus the collineation is perspective. Since by axiom 2.17, its center O does not lie on its axis MN, it is a homology. Finally, since H(PP’, OL), it is a harmonic homology.