## 22474Radical axis - NPCs - Locus (Re: NPCs - Perspective ?)

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• Jun 28, 2014
Let ABC be a triangle, P a point  and A'B'C' the pedal triangle of P.

Denote:

Ab, Ac = the orthogonal projections of A' on AC,AB, resp.
A2, A3 = the reflections of A in Ab, Ac, resp.

Simpler definition of A2, A3:

The circle (A', A'A) intersects AC at A2 and AB at A3

(Oa) = the circumcircle of AA2A3  [Oa = A']
Na = the NPC center of AA2A3

Similarly (Ob), (Oc) and Nb,Nc.

1.  Denote:

Ra = the radical axis of (Ob) and (Oc), Similarly Rb.Rc.
[concurrent at the radical center of the circles]
The parallels to Ra,Rb,Rc through A,B,C, resp. are concurrent (??)

2.  Denote:

Sa = the radical axis of (Nb) and (Nc), Similarly Sb,Sc.
[concurrent at the radical center of the circles]
The parallels to Sa,Sb,Sc though A',B',C', resp. are concurrent (??)

Which is the locus of P such that ABC, NaNbNc are perspective
(or orthologic)?

Antreas

Antreas Hatzipolakis wrote:

Let ABC be a triangle and A'B'C' the orthic triangle.

Denote:

Ab, Ac = the orthogonal projections of A' on AC,AB, resp.
A2, A3 = the reflections of A in Ab, Ac, resp.

Na = the NPC center of AA2A3

Similarly Nb,Nc

I think that the triangles ABC, NaNbNc are perspective.

PS: Is the radical center of (Na),(Nb), (Nc) an interesting point?

APH