Loading ...
Sorry, an error occurred while loading the content.

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

Expand Messages
  • Antreas Hatzipolakis
    Jun 28 11:33 AM
    • 0 Attachment
      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