5.31 Any projectivity that interchanges two distinct points is an involution.
PROOF. Let X be projective to X1 be the given projectivity which interchanges two distinct points A and A1, so that AA1X is projective to A1AX1. By the fundamental theorem 4.12, this projectivity, which interchanges X and X1, is the same as the given projectivity. Since X was arbitrarily chosen, the given projectivity is an involution. Any four collinear points A, A1, B, B1 determine a projectivity AA1B projective to A1AB1, which we now know to be an involution.
The figure below is figure 1.6d(left) but demonstrates this relation. Considering A and B to be one pair and C and D to be the other, we can exchange one point with the other in each set. ABCD is projective through point Q to ZRCW which is projective through point A to QTDW which is projective through point R to BADC.