The polar-chord theorem
If a p-point P lies on the chord AB of a p-conic C, then the polar of P with
respect to C cuts the p-line AB at Q, where (A,B,P,Q) = -1.
Proof
Suppose that A = [a], B = [b], P = [p], Q = [q], and that the
p-conic C
has equation xTMx = 0.
As A,B lie on C, aTMa = bTMb = 0.
As P,Q lie on AB,
(*) p = αa + βb and q = γa + δb.
The polar of P has equation pTMx = 0.
Since Q lies on this polar, pTMq = 0.
Hence (αa + βb)TM(γa + δb) = 0.
Multiplying out, and using the facts that A,B lie on C,
and that aTMb = bTMa, we get
(αδ + βγ)aTMb = 0.
Now the tangent at A has equation aTMx = 0, so that,
as B cannot lie on the tangent (it is on C), aTMb ≠ 0.
Thus αδ + βγ = 0, so βγ/αδ = -1.
But, from (*), this ratio is precisely (A,B,P,Q).
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aTMb is technically a 1x1 matrix,
so is equal to its transpose bTMTa
But M is symmetric, so this is bTMa
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