[APH]
> > Let ABC be a triangle, P = (x:y:z) a point and A'B'C'
> > the pedal triangle of P.
> >
> > Denote:
> >
> > A* := (The Reflection of BC in BP) /\ (The Reflection of BC in CP)
> >
> > B* := (The Reflection of CA in CP) /\ (The Reflection of CA in AP)
> >
> > C* := (The Reflection of AB in AP) /\ (The Reflection of AB in BP)
> >
> > [The triangles A*BC, B*CA, C*AB share the same incenter P]
> >
> > Oa := The Circumcenter of A*BC
> > Ob := The Circumcenter of B*CA
> > Oc := The Circumcenter of C*AB
> >
> > P* := The Point of Concurrence of the Circumcircles
> > of A*BC, B*CA, C*AB
> > [We have seen recently this point in Hyacinthos]
> >
> > La := The Reflection of P*Oa in B'C'
> >
> > Lb := The Reflection of P*Ob in C'A'
> >
> > Lc := The Reflection of P*Oc in A'B'.
> >
> > The Triangles ABC, Triangle bounded by (La,Lb,Lc)
> > are parallelogic.
> >
> > Parallelogic Centers?
> > (one center lies on the circumcircle of ABC)
>
> Francisco told me it is not true in general.
>
> However, it seems it is true for P = O
>
> For P = H, it seems that La,Lb,Lc are concurrent.

Francisco verified that it is true for P = O,H,G,N.
(is it true for all points on the Euler line ?)

He has also computed the coordinates of the parallelogic
centers on the circumcircle.

They are avaliable in a 119 pages pdf file
(1110 KB) I have uploaded to ANOPOLIS FILES.

Probably these coordinates are the hugest ones that ever
have been computed for trianfle centers.
So they have place in the GUINNESS RECORDS BOOK !!!

See:

http://tech.groups.yahoo.com/group/Anopolis/message/126

Thanks once again to Francisco

Antreas