# Coxeter- Theorem 5.31

- Author:
- AndrewLyon

- Topic:
- Geometry

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.*