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Circumcenter and Incenter

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  • mbabelian
    Dear Friends Let ABC and O and I are its circumcenter and incenter,respectively. Let M be a point on the BC and U and V are the incenters of triangles ABM and
    Message 1 of 4 , Apr 2, 2012
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      Dear Friends

      Let ABC and O and I are its circumcenter and incenter,respectively.
      Let M be a point on the BC and U and V are the incenters of triangles ABM and ACM. Let T the circumcenter of triangle AUV.
      prove that OT and AI are parallel if and only if M = midpoint of BC

      With Regards.
    • Chandan Banerjee
      Suppose, AN is the isogonal conjugate of AM wrt angle BAC where N lies on the circumcircle of ABC. External angle-bisector of angle BAC intersects BC at X.
      Message 2 of 4 , Apr 2, 2012
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        Suppose, AN is the isogonal conjugate of AM wrt \angle BAC where N lies on the circumcircle of ABC.
        External angle-bisector of \angle BAC intersects BC at X.
        Under inversion wrt A followed by a reflection on the angle-bisector of BAC, suppose N goes to M.
        Then U',V' be the images of U,V.
        Clearly, U',V' are the A-excenters of ANC and ANB.
        K=AU'\cap NC
        L=AV'\cap NB
        P,Q be the incenters of ANC and ANB. Note that, PQ,KL,U'V' are concurrent. Also KL intersects BC at X.
        And from AI||OT we get that U'V' intersects BC at X too.
        So PQ,KL,U'V',BC are concurrent.
        AU'\cap BC=J, AV'\cap BC=H
        So (AJ;KP)=(AH;LQ). So CP,BQ meet on AN.
        So BN/NC=AB/AC. So AN is the A-symmedian of ABC.
        So M is midpoint of BC.
      • Jean-Louis Ayme
        Dear Hyacinthists, Dear mbabelian,   it would be nice to mention the origin of this problem
        Message 3 of 4 , Apr 2, 2012
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          Dear Hyacinthists,
          Dear mbabelian,
           
          it would be nice to mention the origin of this problem
          http://www.artofproblemsolving.com/Forum/viewtopic.php?f=46&t=472759
          the author being Jean-Louis Ayme
           
          Sincerely
          Jean-Louis


          ________________________________
          De : mbabelian <mbabelian@...>
          À : Hyacinthos@yahoogroups.com
          Envoyé le : Lundi 2 avril 2012 21h43
          Objet : [EMHL] Circumcenter and Incenter


           
          Dear Friends

          Let ABC and O and I are its circumcenter and incenter,respectively.
          Let M be a point on the BC and U and V are the incenters of triangles ABM and ACM. Let T the circumcenter of triangle AUV.
          prove that OT and AI are parallel if and only if M = midpoint of BC

          With Regards.




          [Non-text portions of this message have been removed]
        • mbabelian
          Dear Friends, Dear Jean-Louis I just found it in a weblog and it didn t mention the author of it. sorry. Thank you for mentioning its origin. With Regards.
          Message 4 of 4 , Apr 2, 2012
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            Dear Friends,
            Dear Jean-Louis
            I just found it in a weblog and it didn't mention the author of it. sorry.
            Thank you for mentioning its origin.
            With Regards.

            --- In Hyacinthos@yahoogroups.com, Jean-Louis Ayme <jeanlouisayme@...> wrote:
            >
            > Dear Hyacinthists,
            > Dear mbabelian,
            >  
            > it would be nice to mention the origin of this problem
            > http://www.artofproblemsolving.com/Forum/viewtopic.php?f=46&t=472759
            > the author being Jean-Louis Ayme
            >  
            > Sincerely
            > Jean-Louis
            >
            >
            > ________________________________
            > De : mbabelian <mbabelian@...>
            > À : Hyacinthos@yahoogroups.com
            > Envoyé le : Lundi 2 avril 2012 21h43
            > Objet : [EMHL] Circumcenter and Incenter
            >
            >
            >  
            > Dear Friends
            >
            > Let ABC and O and I are its circumcenter and incenter,respectively.
            > Let M be a point on the BC and U and V are the incenters of triangles ABM and ACM. Let T the circumcenter of triangle AUV.
            > prove that OT and AI are parallel if and only if M = midpoint of BC
            >
            > With Regards.
            >
            >
            >
            >
            > [Non-text portions of this message have been removed]
            >
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