HYACINTHOS 21942

 

 [Antreas P. Hatzipolakis]:

> > Let ABC be a triangle, P, P* two isogonal conjugate points.
> >
> > Denote:
> >
> > Ra = radical axis of (NPC_PBC), (NPC_P*BC)
> >
> > Rb = radical axis of (NPC_PCA), (NPC_P*CA)
> >
> > Rc = radical axis of (NPC_PAB), (NPC_P*AB)
> >
> > Which is the locus of P such that Ra,Rb,Rc are concurrent?
> > The entire plane?
>
> I wrote this without making a figure. I had in mind
> the points P,P* = O,H: the triangles HBC,HCA, HAB
> share the same NPC, the NPC of ABC, and the NPCs of OBC, OCA, OAB
> concur at the Poncelet point of O wrt ABC, lying on the NPC of ABC.
> Therefore the radical axes in question are concurrent for
> that points.
> Now I make a figure with P,P* = G,K and it seems that
> the radical axes are again concurrent.
>
> See the figure:
>
> http://anthrakitis.blogspot.gr/2013/04/radical-axes-of-npcs.html
>
> Is it true? And if yes, which is the point of concurrence?
>
> In general???
>
> APH

[Randy Hutson]:

Antreas,

I tried various specific pairs of isogonal points, and finally for an
arbitrarily random pair, and it appears the locus is the entire plane. This
does provide an interesting mapping. Some specifics for the concurrence
point for (P,P*):

(G,K): non-ETC -0.956613251489256, which is also the Hyacinthos #16741/16782
homothetic center for line X(2)X(6), and the centroid of (degenerate) pedal
triangle of X(111).

(O,H): X(125)

(N,X54): non-ETC 4.975239945739461

(X7,X55): non-ETC 3.097837435698617

(PU(1)): non-ETC 4.098938269094193, which is the center of conic
{A,B,C,X(99),PU(37)}, complement of X(1916), and anticomplement of X(2023),
and lies on lines 2,694 3,76.

(foci of Steiner inellipse): non-ETC 0.632166489381459, which is the center
of the circumconic through the isotomic conjugates of the foci of the
Steiner inellipse; also the crosssum of X(6) and X(1380), the crosspoint of
X(2) and X(3414); also the complement of the trilinear pole of major axis of
the Steiner eliipses (line X(2)X(1341)); lies on line X(2)X(6) and on the
Steiner inellipse.

(foci of orthic inconic): non-ETC 4.706821577388139, which is the isotomic
conjugate of the polar conjugate of X(1313).

I think this is certainly worth exploring further.

Best regards,
Randy

[Cesar Lozada]:

The trilinear transformation of P( u : v : w ) is :



( f(u, v, w, a, b, c, A, B, C) : f(v, w, u, b, c, a, B, C, A) : f( w, u, v,
c, a, b, C, A, B) )



where



f(u, v, w, a, b, c, A, B, C) = u*(v^2 - w^2)*[u*( v^2 - w^2) - v*(w^2 -
u^2)*cos(C) - w*( u^2 - v^2)*cos(B) ] /a



whose inverse doesn’t seem to be easily calculable.



Regards

Cesar Lozada