If 4 collinear p-points embed as non-ideal points on a plane Π, then the
cross-ratio 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 cross-ratio is -1, when we say
To show how information from a projective figure can be used to obtain
A mid-point theorem
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 mid-point 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 PQ||N,
To see how such a cross-ratio may arise in practice, we will look at some
projective cross ratios