On Fri, 16 Jun 2000, Paul Yiu wrote [edited]:

>

> Let P be a fixed point, and L a given line with tripole L*.

> Then P/L is the conic through the traces of P and L*.

It seems to me that "the conic through the traces of P and Q"

deserves a name! Let's call it [P,Q] for the moment, so that

your theorem asserts that P/Q* = [P,Q], which is symmetric in

P and Q, so you've proved that P/Q* = Q/P* - wonderful!

This has, I think, some notational consequences. It's natural to

use 1/P for the tripolar, in view of the coordinate form:

tripolar of the point (X:Y:Z) is the line (1/X:1/Y:1/Z|).

In this notation, your result becomes

P/(1/Q) = Q/(1/P) is a symmetric "product" [P,Q]

which might almost be expected! I wonder if there's any more

"algebra" of this type?

> Corollary:

>

> Let P be a fixed point. Then P/P* is the inscribed conic with

> Brianchon point P. The center of the conic is the inferior of the

> isotomic conjugate of P.

I think your "Brianchon point" of an inconic is the same as my

"perspector" of it? I used to use various names for these notable

invariant points of inconics and circumconics until I noticed only

last year the important fact that ABC is in perspective with its

dual triangle with respect to ANY conic, not just in- and circum- ones,

and so switched to the term "perspector" for all cases.

Regards, John Conway