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• Let ABC be a triangle, P a point on the Euler line, and Pa,Pb,Pc its reflections in the altitudes AA , BB , CC , resp. Which is the locus of the NPC center of
Message 1 of 4 , Jul 8, 2009
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Let ABC be a triangle, P a point on the Euler line,
and Pa,Pb,Pc its reflections in the altitudes
AA', BB', CC', resp.

Which is the locus of the NPC center of PaPbPc as
P moves on the Euler Line of ABC?

APH

[APH]
> > > Let ABC be a triangle and A'B'C' the cevian triangle
> > > of H (orthic triabgle).
> > >
> > > Denote:
> > >
> > > (Na), (Nb), (Nc) := The Reflections of (N) = the NPC of ABC,
> > > in AA', BB', CC', resp.
> > >
> >
> > >
> > > About the triangle NaNbNc:
> > >
> >
> > Let (Q) be the circumcircle of NaNbNc
> > and Sa, Sb, Sc the radical axes of [(Q),(Na)], [(Q),(Nb)], [(Q),(Nc)], resp.
> >
> > The Triangles ABC, Triangle bounded by (Sa,Sb,Sc)
> > are orthologic.
> >
> > The Triangles A'B'C', Triangle bounded by (Sa,Sb,Sc)
> > are orthologic, with one orth. center on (N) = NPC of ABC.
> >
> > Centers?
>
> The circumcenter of NaNbNc is H.
> Since Sa,Sb,Sc are perpendiculars to HNa, HNb, Hnc resp.,
> the problem is equivalent to:
>
> The parallels from A,B,C to HNa, HNb, HNc, resp.
> are concurrent.
>
> The parallels through A',B',C' to HNa, HNb, HNc are
> concurrent on a point on the NPC of ABC.
>
> In fact, it is true for any point N.
>
> That is:
>
> Let ABC be a triangle, A'B'C' its orthic triangle
> and P a point.
>
> Let Pa, Pb, Pc be the reflections of P in AH, BH, CH, resp.
>
> 1. The parallels to HPa, HPb, HPc through A,B,C, resp.
> are concurrent
>
> 2. The parallels to HPa, HPb, HPc through A',B',C'
> are concurrent on the NPC of ABC.
>
> And if we take ABC instead of the orthic A'B'C':
>
> Let ABC be a triangle and P a point.
>
> Let Pa, Pb, Pc be the reflections of P in AI, BI, CI,
> resp.
>
> 1. The parallels to IPa, IPb, IPc trough the vertices
> of the excentral triangle Ia, Ib, Ic, resp. are concurrent.
>
> 2. The parallels to IPa, IPb, IPc through A,B,C, resp.
> are concurrent on the circumcircle of ABC.
>
>
> Points??
>
>
> Antreas
>
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