Basic results on L-conjugates
Suppose that C is a p-conic, and L a p-line not tangent to C.
Let P be the pole of L with respect to C.
(1) If M is an L-diameter, then ML is an L-diameter.
(2) If N = ML, then M = NL.
(3) M = ML if and only if M and L meet on C.
proof
Suppose that M is an L-diameter and that L and M meet in Q.
(1) By definition, ML is the polar of Q.
Now, Q lies on the L the polar of P.
By La Hire's Theorem, P lies on the polar of Q, i.e. on ML.
(2) Suppose that N = ML and L meet in R.
Then, by definition, NL is the polar of R.
Now, R lies on ML is the polar of Q, so, by La Hire's Theorem, Q is on the polar of R.
Thus, Q is on NL. By (1), applied to N, P is on NL.
Hence NL
is the p-line PQ.
But P,Q lie on M, so NL = M.
(3) Since P and Q lie on M, M is defined by P and Q.
By (1), P lies on ML, so
M = ML
if and only if Q is on ML.
Now, ML is the polar of Q, so Q is on ML if and only if Q is on C,
by the Lemma.
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