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  • Jean-Louis Ayme
    Dear Jean-Pierre and Mathlinkers, working on a generalization of a Nagel s result, I found in the archive of Hyacinthos the message # 19640 Dear Chris [JP]
    Message 1 of 2 , Feb 24 5:48 AM
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      Dear Jean-Pierre and Mathlinkers,
      working on a generalization of a Nagel's result, I found in the archive of Hyacinthos the message # 19640
      Dear Chris
      [JP]
      > > Consider a quadrilateral A_1,A_2,A_3,A_4
      > > For k =1,2,3,4, T_k is the triangle with vertices the A_i except A_k; O_k is
      the circumcenter of T_k, (O_k) the circumcircle and B_k the isogonal conjugate
      of A_k wrt T_k
      > > Then the inverse of B_k in (O_k) doesn't depend on k and this point M is the
      center of the homothecy mapping O_1,O_2,O_3,O_4 to B_1,B_2,B_3,B_4.
      > > Is there a special name for this point M? Do you know some references?
      [Chris]
      > I do not know a special name for point M.
      > However I noticed this. Maybe you know it already.
      > Let T(A_1,A_2,A_3,A_4) = Transform A_1,A_2,A_3,A_4 --> O_1,O_2,O_3,O_4.
      > Then T^2(A_1,A_2,A_3,A_4) produces a quadrilateral homethetic with
      A_1,A_2,A_3,A_4 only rotated 180 degrees. Again Center of Homothecy = M.
      > T^4(A_1,A_2,A_3,A_4) produces a quadrilateral homothetic and with same
      orientation as A_1,A_2,A_3,A_4.

      Thank you for your nice remark.
      In fact, if O_1' is the circumcenter of O_2O_3O_4,...,
      the same homothecy maps A_i to O_i' and B_i to O_i
      The point M is characterized by the angular relations
      <A_iMA_j = <A_iA_kA_j +<A_iA_lA_j (oriented angles modulo Pi) where (i,j,k,l) is
      any permutation of (1,2,3,4)
       
      Ma question is: how can we prouve the last angular relations? Is there a simple way to prouve it? I try to discover a synthetic proof without success.
      Sincerely
      Jean-Louis

      [Non-text portions of this message have been removed]
    • jpehrmfr
      Dear Jean-Louis and other Hyacinthists this follows from two facts : if P,P are inverse in the circumcircle of ABC, then
      Message 2 of 2 , Feb 25 5:49 AM
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        Dear Jean-Louis and other Hyacinthists
        this follows from two facts :
        if P,P' are inverse in the circumcircle of ABC, then
        <BPC + <BP'C = 2<BAC
        if P,P* are isogonal conjugates wrt ABC, then
        <BPC + <BP*C = <BAC
        Kind regards. Jean-Pierre

        --- In Hyacinthos@yahoogroups.com, Jean-Louis Ayme <jeanlouisayme@...> wrote:
        >
        > Dear Jean-Pierre and Mathlinkers,
        > working on a generalization of a Nagel's result, I found in the archive of Hyacinthos the message # 19640
        > Dear Chris
        > [JP]
        > > > Consider a quadrilateral A_1,A_2,A_3,A_4
        > > > For k =1,2,3,4, T_k is the triangle with vertices the A_i except A_k; O_k is
        > the circumcenter of T_k, (O_k) the circumcircle and B_k the isogonal conjugate
        > of A_k wrt T_k
        > > > Then the inverse of B_k in (O_k) doesn't depend on k and this point M is the
        > center of the homothecy mapping O_1,O_2,O_3,O_4 to B_1,B_2,B_3,B_4.
        > > > Is there a special name for this point M? Do you know some references?
        > [Chris]
        > > I do not know a special name for point M.
        > > However I noticed this. Maybe you know it already.
        > > Let T(A_1,A_2,A_3,A_4) = Transform A_1,A_2,A_3,A_4 --> O_1,O_2,O_3,O_4.
        > > Then T^2(A_1,A_2,A_3,A_4) produces a quadrilateral homethetic with
        > A_1,A_2,A_3,A_4 only rotated 180 degrees. Again Center of Homothecy = M.
        > > T^4(A_1,A_2,A_3,A_4) produces a quadrilateral homothetic and with same
        > orientation as A_1,A_2,A_3,A_4.
        >
        > Thank you for your nice remark.
        > In fact, if O_1' is the circumcenter of O_2O_3O_4,...,
        > the same homothecy maps A_i to O_i' and B_i to O_i
        > The point M is characterized by the angular relations
        > <A_iMA_j = <A_iA_kA_j +<A_iA_lA_j (oriented angles modulo Pi) where (i,j,k,l) is
        > any permutation of (1,2,3,4)
        >  
        > Ma question is: how can we prouve the last angular relations? Is there a simple way to prouve it? I try to discover a synthetic proof without success.
        > Sincerely
        > Jean-Louis
        >
        > [Non-text portions of this message have been removed]
        >
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