If 4 collinear ppoints embed as nonideal points on a plane Π, then the crossratio is a ratio of signed ratios in Π (see the embedding theorem)
The Greeks studied this, thinking of the quantity as defined by a pencil
A particularly interesting case is when the crossratio is 1, when we say
To show how information from a projective figure can be used to obtain
A midpoint theorem


Proof From case (5) of the the embedding theorem, we have (A,B,C,D) = C'A'/C'B'. Thus C'A'/C'B' = 1, so C' is the midpoint of A'B'. We can rephrase this as a euclidean result:
If (K,L,M,N) is a harmonic pencil and P,Q lie on K,L such that PQN,
To see how such a crossratio may arise in practice, we will look at some

