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22158Re: [Quadri-Figures-Group] Re: A quadrangle of quadrangle

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  • Antreas Hatzipolakis
    Apr 20 4:56 AM
    • 0 Attachment


       

      [AH] Let D' be the point of concurrence of the above radical axes in triangle ABC and point D

      and similarly C', B',A'. Which properties has the quadrangle A'B'C'D'? Are the lines AA', BB', CC', DD' concurrent?

       

      [Chris]

      However alas your P-NPC-Radical-Axis-Triangle Transformation doesn't produce a Quadrangle Perspector.

      Neither for the 1st nor the 2nd generation.


      *************************************

      Dear Chris,

      Thank you!

      So, naturally we have to return to triangle geometry and ask for locus!

      That is:

      Let ABC be a triangle and P a point.

      Denote:
      Ap, Bp, Cp = the isogonal conjugates of A,B,C, resp. wrt triangles
      PBC, PCA, PAB, resp.
      rA1 = the radical axis of NPC_ABC and NPC_ApBC
      rA2 = the radical axis of NPC_APC and NPC_ApPC
      rA3 = the radical axis of NPC_APB and NPC_ApPB
      A' = the point of concurrence of rA1, rA2, rA3.
      Similarly (cyclically) B',C'

      Which is the locus of P such that ABC, A'B'C' are perspective?


      Antreas