## Cevian triangles similar to ABC

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• Dear Hyacinthists, If ABC is obtusangle and D the line isotomic conjugate of the circumcircle, D intersects the circumcircle in two points V and W ( W is the
Message 1 of 32 , May 31, 2000
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Dear Hyacinthists,
If ABC is obtusangle and D the line isotomic conjugate of the circumcircle,
D intersects the circumcircle in two points V and W ( W is the isotomic
conjugate of V and the reflection of V w.r.t. the Euler line).
V and W have a very nice property : their Cevian triangles are directly
similar to ABC (the two centers of similitude are the common points of the
circumcircle and the nine-points circle ); more over their Cevian triangles
are congruent.

Has any one some references about those points V and W?
I think they should have a lot of other nice properties.

Friendly from France. Jean-Pierre
• ... 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
Message 32 of 32 , Jun 16, 2000
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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
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