# Coxeter- Theorem 5.32

- Author:
- AndrewLyon

- Topic:
- Geometry

5.32

*An involution is determined by any two of its pairs.*Following from theorem 5.31, it is convenient to denote the involution*AA1B*projective to*A1AB1*by*(AA1)(BB1)*or*(A1A)(BB1)*, or*(BB1)(AA1)*, and so forth. This notation reminas valid when*B1*coincides with*B*; in other words, the involution*AA1B*projective to*A1AB*, for which*B*is invariant, may be denoted by*(AA1)(BB)*. If*(AD)(BE)(CF)*, as in Figure 2.4A (reproduced and provided), we can combine the projectivity*AECF*projective to*BDCF*of 5.11 with the involution*(BD)(CF)*to obtain*AECF*projective to*BDCF*which is projective to*DBFC*which shows there there is a projectivity in which*AECF*projective to*DBFC*. Since this interchanges*C*and*F*, it is an involution, namely*(BE)(CF)*or*(CF)(AD)*or*(AD)(BE)*; all equivalent statements. Thus the quadrangular relation*(AD)(BE)(CF)*is eqquivalent to the statement that the projectivity*ABC*projective to*DEF*is an involution, or that*ABCDEF*projective to*DEFABC*.