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Two tangent circles

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  • Jean-Louis Ayme
    Dear Hyacinthists, during resolving another problem, I have found this one... Let ABC be a right triangle at A and O the midpoint of BC. Prove that the circle
    Message 1 of 5 , Sep 21, 2010
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      Dear Hyacinthists,
      during resolving another problem, I have found this one...
      Let ABC be a right triangle at A and O the midpoint of BC.
      Prove that the circle with diameter AO is tangent to the incircle.
      Sincerely
      Jean-Louis




      [Non-text portions of this message have been removed]
    • Jean-Louis Ayme
      Dear Hyacinthists, sorry for the simplicity of this problem!!!! Sincerely Jean-Louis ... De: Jean-Louis Ayme Objet: [EMHL] Two tangent
      Message 2 of 5 , Sep 21, 2010
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        Dear Hyacinthists,
        sorry for the simplicity of this problem!!!!
        Sincerely
        Jean-Louis

        --- En date de : Mer 22.9.10, Jean-Louis Ayme <jeanlouisayme@...> a écrit :


        De: Jean-Louis Ayme <jeanlouisayme@...>
        Objet: [EMHL] Two tangent circles
        À: Hyacinthos@yahoogroups.com
        Date: Mercredi 22 septembre 2010, 7h06


         



        Dear Hyacinthists,
        during resolving another problem, I have found this one...
        Let ABC be a right triangle at A and O the midpoint of BC.
        Prove that the circle with diameter AO is tangent to the incircle.
        Sincerely
        Jean-Louis

        [Non-text portions of this message have been removed]











        [Non-text portions of this message have been removed]
      • Vijaya Prasad Nalluri
        Circles (N;a/4) and (I;r) touch because  r = s - a = (b + c - a) /2; b + c = 2r + a AI = r / cos pi/4 = r sqrt2,  AN = a/4,  angle IAN = pi/4 ~ C,  cos
        Message 3 of 5 , Sep 22, 2010
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          Circles (N;a/4) and (I;r) touch because 
          r = s - a = (b + c - a) /2; b + c = 2r + a
          AI = r / cos pi/4 = r sqrt2, 
          AN = a/4, 
          angle IAN = pi/4 ~ C, 
          cos (IAN) =  (cos C + Sin C) / (sqrt 2)              =  (b + c) / a (sqrt 2)              =  (2r + a) / a (sqrt 2)
          I NI I^2 = I AN I^2 + I AI I^2 - 2. I AN I. I AI I. cos (IAN)            = (a/4)^2 + 2r^2 - 2. (a/4). (r sqrt2). (2r + a) / a (sqrt 2)           = (a/4)^2 + 2(r^2) - r(2r + a) / 2           = (a/4)^2 + r^2 - ar / 2           = (r ~ a/4)^2 etc
          Sincerely,Vijayaprasad.
          --- On Wed, 22/9/10, Jean-Louis Ayme <jeanlouisayme@...> wrote:

          From: Jean-Louis Ayme <jeanlouisayme@...>
          Subject: Re : [EMHL] Two tangent circles
          To: Hyacinthos@yahoogroups.com
          Date: Wednesday, 22 September, 2010, 10:56 AM
















           









          Dear Hyacinthists,

          sorry for the simplicity of this problem!!!!

          Sincerely

          Jean-Louis



          --- En date de : Mer 22.9.10, Jean-Louis Ayme <jeanlouisayme@...> a écrit :



          De: Jean-Louis Ayme <jeanlouisayme@...>

          Objet: [EMHL] Two tangent circles

          À: Hyacinthos@yahoogroups.com

          Date: Mercredi 22 septembre 2010, 7h06



           



          Dear Hyacinthists,

          during resolving another problem, I have found this one...

          Let ABC be a right triangle at A and O the midpoint of BC.

          Prove that the circle with diameter AO is tangent to the incircle.

          Sincerely

          Jean-Louis



          [Non-text portions of this message have been removed]



          [Non-text portions of this message have been removed]





























          [Non-text portions of this message have been removed]
        • bbblow
          The circle with AO s diameter is nine-point-circle of ABC (hence by feuerbach s thm the problem becomes obvious) (This was sent to the problem setter via email
          Message 4 of 5 , Sep 23, 2010
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            The circle with AO s diameter is nine-point-circle of ABC (hence by feuerbach's thm the problem becomes obvious)

            (This was sent to the problem setter via email but wasn't uploaded on hyacinthos so I'm posting again)
          • jeanlouisayme
            Dear Mathlinkers, an article IN ENGLISH concerning a circle tangent to the incircle has been put on my website http://perso.orange.fr/jl.ayme vol. 6 A circle
            Message 5 of 5 , Sep 24, 2010
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              Dear Mathlinkers,
              an article IN ENGLISH concerning a circle tangent to the incircle has been put on my website
              http://perso.orange.fr/jl.ayme vol. 6 A circle tangent to the incircle
              Sincerely
              Jean-Louis


              --- In Hyacinthos@yahoogroups.com, Jean-Louis Ayme <jeanlouisayme@...> wrote:
              >
              > Dear Hyacinthists,
              > during resolving another problem, I have found this one...
              > Let ABC be a right triangle at A and O the midpoint of BC.
              > Prove that the circle with diameter AO is tangent to the incircle.
              > Sincerely
              > Jean-Louis
              >
              >
              >
              >
              > [Non-text portions of this message have been removed]
              >
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