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22475Another Inversion Th. ? [Re: Generalization of P. Moses Circle (Re: Trilinear poles)]

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  • Antreas Hatzipolakis
    Jun 28, 2014
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      Another Inversion Theorem (?)

      Let (A), (B), (C) be three circles tangent all externally or internally
      a circle Q at A',B',C', resp.

      Denote:

      Ab = the inversion of A' in (B)
      Ac = the inversion of A' in (C)
      Bc = the inversion of B' in (C)
      Ba = the inversion of B' in (A)
      Ca = the inversion of C' in (A)
      Cb = the inversion of C' in (B)

      Ma = the midpoint of BaCa
      Mb = the midpoint of CbAb
      Mc = the midpoint of AcBc

      The lines AMa, BMb, CMc are concurrent ??

      If true, then we can apply it to Circumcircle and three Lucas Circles,
      to NPC and three excircles etc

      Antreas


      [APH]

      Let (A), (B), (C) be three externally tangent circles.
      The [Apollonian] circle tangent to (A),(B),(C) internally
      touches them at A',B',C', resp.

      Denote:

      Ab = the inversion of A' in (B)
      Ac = the inversion of A' in (C)
      Bc = the inversion of B' in (C)
      Ba = the inversion of B' in (A)
      Ca = the inversion of C' in (A)
      Cb = the inversion of C' in (B)

      Are the six points Ab,Ac.Bc,Ba,Bc,Ca,Cb always concyclic?

      In my figure they are, and the center of the circle lies
      on the line joining the circumcenters of ABC and A'B'C'.
       
      Is it true? And if yes, is it already known? (references?)

      Antreas