5.31 [i]Any projectivity that interchanges two distinct points is an involution.[/i] PROOF. Let[i] X[/i] be projective to[i] X1[/i] be the given projectivity which interchanges two distinct points[i] A[/i] and [i]A1[/i], so that [i]AA1X[/i] is projective to [i]A1AX1[/i]. 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 [i]A, A1, B, B1[/i] determine a projectivity[i] AA1B[/i] projective to [i]A1AB1[/i], 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. [i]ABCD[/i] is projective through point [i]Q[/i] to [i]ZRCW[/i] which is projective through point [i]A[/i] to [i]QTDW[/i] which is projective through point [i]R[/i] to [i]BADC.[/i]