# Coxeter- Figure 5.1A

- Author:
- AndrewLyon

- Topic:
- Geometry

5.11 The two statements

*AECF*projective to*BDCF*and*(AD)(BE)(CF)*are equivalent, not only when*C*and*F*are distinct, but also when they coincide. Since the statement*AECF*projective to*BDCF*involves*C*and*F*symmetrically, the statement*(AD)(BE)(CF)*is equivalent to*(AD)(BE)(FC)*, and similarly to*(AD)(EB)(FC)*and to*(DA)(EB)(FC)*. This is remarkable because, when the quadrangular set is derived from the quadrangle, the two triads*ABC*and*DEF*arise differently: the first from three sides with a common vertex, and the second from three that form a triangle. It is interesting that, whereas one way of matching two quadrangles (Figure 2.4b) uses only Desargues's theorem, the other needs the fundamental theorem.