Coxeter- Figure 5.1B
- 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.