Dear Bernard and Steve

> >> [JP] consider the rectangular hyperbola through G, O, K

> >> (symedian), X[110).

> >> His center is the midpoint of GX[110] and the hyperbola goes

through

> >> the infinite points of the Jerabek hyperbola, X[154], X[354], X

> >> [392], X

> >> [1201],...

> >> Consider a common point P (not X[110]) of the hyperbola with the

> >> circumcircle;

>

> These 3 points P lie on the Thomson cubic.

I don't think so

> >> let M be the homothetic of P in (O, -3) and A'B'C' the

> >> pedal triangle of M.

> >> Then AA', BB', CC' are parallel.

> >> Any explanation?

> there's a close analogy with the bottom of my web page on the

Darboux

> cubic.

>

> http://perso.wanadoo.fr/bernard.gibert/Exemples/k004.html

Yes, indeed. These three points are UVW in your Darboux page. many

thanks.

It is quite interesting to see that these points are the only points

P in the plane such as, if A'B'C' is the pedal triangle of P, the

lines AA', BB', CC' are parallel (of course they are parallel to the

asymptots of Lucas - or Thomson -)

Friendly. Jean-Pierre