the cross-ratio theorem for chords of a projective conic
Suppose A, B are distinct p-points on the projective conic C,
and that P,Q are distinct p-points on AB, but not on C. Then
P,Q both lie inside C or both lie outside C
if and only if (A,B,P,Q) is positive.
proof
Suppose that C has equation f(x) = xTMx = 0.
Let A =[a], B =[b], P =[p], Q =[q].
As A, B lie on C, f(a) = f(b) = 0.
As B ≠ A, B is not on the polar of (i.e the tangent at) A,
so k = aTMb ≠ 0.
As P, Q are on AB, there exist non-zero α,β,γ,δ such that
p = αa+βb, and q = γa+δb.
Then, replacing p by the above, and multiplying out,
f(p) = pTMp = α2f(a) + 2αβk + β2f(b) = 2αβk,
since f(a) = f(b) = 0.
Likewise, f(q) = 2γδk.
By the the algebraic interior-exterior theorem
P, Q both lie inside C, or both lie outside C
if and only if f(p), f(q) have the same sign,
i.e 2αβk and 2γδk have the same sign and hence
αβ and γδ have the same sign. It follows that
(A,B,C,D) = βγ/αδ is positive only in these cases.
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