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22473Re: Generalization of P. Moses Circle (Re: Trilinear poles) [1 Attachment]

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  • Antreas Hatzipolakis
    Jun 28, 2014
    • 0 Attachment



      [APH]

      I had written my message without drawing a figure. Now I have drawn one.

      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


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

      Solution by Telv Cohl in "Simple Mathematical Idea" FB Group

      Inverse with center A'
      P1 = the image of P under this inversion
      [Apollonian] circle = (K)

      easy to see
      (K)1 is the line which is parallel to (A)1
      and (B)1 (C)1 are two congruent circle
      (tangent to (K)1 and (A)1)
      (B)1 is tangent to (K)1 at B'1
      (C)1 is tangent to (K)1 at C'1
      Ba1 is the reflection point of B'1 wrt (A)1 ... (1)
      Ca1 is the reflection point of C'1 wrt (A)1 ... (2)

      Since A'1 is the point at infinity
      so Ab1 Ac1 is the center of (B)1 (C)1 respectively ... (3)
      From (1) (2) (3) we get
      Ba1Ab1Ac1Ca1 is rectangle and Ba1 Ab1 Ac1 Ca1 are concyclic
      so Ab Ac Ba Ca are concyclic ... (4)

      similarity
      Ba Bc Ab Cb are concyclic ... (5)
      Cb Ca Ac Bc are concyclic ... (6)

      From (4) (5) (6) and David's theorem
      we deduce that Ab Ac Ba Bc Ca Cb are concyclic
      Q.E.D


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

      Now the question is about the general case of three non-tangent
      circles tangent internally or externally a circle.

      That is:

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

      Denote:

      Ab, Ac = the reversions of A' in (B),(C), resp.
      Similarly Bc,Ba,Ca,Cb,

      Under what conditions the six points Ab,Ac, Bc, Ba, Ca, Cb
      are lying on a conic?

      Application to triangle:

      We take as (A), (B), (C) the three excircles, tangent to NPC, and
      as A', B', C' the ex-Feuerbach points. That Is:

      Let Fa, Fb, Fc be the three ex-Feuerbach poinrs.

      Denote:

      Ab = the inversion of Fa in the excircle (Ib)
      Ac = the inversion of Fa in the excircle (Ic)

      Bc = the inversion of Fb in the excircle (Ic)
      Ba = the inversion of Fb in the excircle (Ia)

      Ca = the inversion of Fc in the excircle (Ia)
      Cb = the inversion of Fc in the excircle (Ib)

      Are the six points Ab,Ac,Bc,Ba,Ca,Cb lying on a conic?

      Antreas





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