Finding the set of all triangles satisfying the given condition

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Let $S$ be the set of all triangles $ABC$ for which



$ 5left( dfrac1AP + dfrac1BQ + dfrac1CR right) - dfrac3min AP, BQ, CR = dfrac6r,$
where $r$ is the inradius and $P, Q, R$ are the points of tangency of the incircle with sides $AB, BC, CA,$ respectively. How do I Prove that all triangles in $S$ are isosceles and similar to one another.



After taking $s-a=x$, $s-b=y$, $s-c=z$ and assuming WLWOG that $x le y le z$ after much of grueling work I got an equation



$0 = 4y^2z^2 + 25x^2y^2 + 25x^2z^2 - 16xy^2z - 16xyz^2 + 14x^2yz$



How do I solve this equation to get the required result?










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    Let $S$ be the set of all triangles $ABC$ for which



    $ 5left( dfrac1AP + dfrac1BQ + dfrac1CR right) - dfrac3min AP, BQ, CR = dfrac6r,$
    where $r$ is the inradius and $P, Q, R$ are the points of tangency of the incircle with sides $AB, BC, CA,$ respectively. How do I Prove that all triangles in $S$ are isosceles and similar to one another.



    After taking $s-a=x$, $s-b=y$, $s-c=z$ and assuming WLWOG that $x le y le z$ after much of grueling work I got an equation



    $0 = 4y^2z^2 + 25x^2y^2 + 25x^2z^2 - 16xy^2z - 16xyz^2 + 14x^2yz$



    How do I solve this equation to get the required result?










    share|cite|improve this question

























      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      Let $S$ be the set of all triangles $ABC$ for which



      $ 5left( dfrac1AP + dfrac1BQ + dfrac1CR right) - dfrac3min AP, BQ, CR = dfrac6r,$
      where $r$ is the inradius and $P, Q, R$ are the points of tangency of the incircle with sides $AB, BC, CA,$ respectively. How do I Prove that all triangles in $S$ are isosceles and similar to one another.



      After taking $s-a=x$, $s-b=y$, $s-c=z$ and assuming WLWOG that $x le y le z$ after much of grueling work I got an equation



      $0 = 4y^2z^2 + 25x^2y^2 + 25x^2z^2 - 16xy^2z - 16xyz^2 + 14x^2yz$



      How do I solve this equation to get the required result?










      share|cite|improve this question















      Let $S$ be the set of all triangles $ABC$ for which



      $ 5left( dfrac1AP + dfrac1BQ + dfrac1CR right) - dfrac3min AP, BQ, CR = dfrac6r,$
      where $r$ is the inradius and $P, Q, R$ are the points of tangency of the incircle with sides $AB, BC, CA,$ respectively. How do I Prove that all triangles in $S$ are isosceles and similar to one another.



      After taking $s-a=x$, $s-b=y$, $s-c=z$ and assuming WLWOG that $x le y le z$ after much of grueling work I got an equation



      $0 = 4y^2z^2 + 25x^2y^2 + 25x^2z^2 - 16xy^2z - 16xyz^2 + 14x^2yz$



      How do I solve this equation to get the required result?







      geometry triangle






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      edited Sep 10 at 13:26

























      asked Sep 7 at 9:03









      saisanjeev

      517212




      517212




















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          The question you wrote is wrong, it should be $$5left( dfrac1AP + dfrac1BQ + dfrac1CR right) - dfrac3min AP, BQ, CR = dfrac6r,\ and not left( dfrac1AP + dfrac1BQ + dfrac1CR right) - dfrac3min AP, BQ, CR = dfrac6r, $$
          This is your answer



          I didn't write the answer cause it would take much time , sorry for that.






          share|cite|improve this answer




















          • Can you please provide the full solution as google books is not allowing me to view the complete solution
            – saisanjeev
            Sep 10 at 13:28










          • I just searched for it and posted the link , ok then please wait and let me try the solution myself . I am just a 12 th class student.
            – Subhajit Halder
            Sep 10 at 13:32










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          1 Answer
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          active

          oldest

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          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes








          up vote
          0
          down vote













          The question you wrote is wrong, it should be $$5left( dfrac1AP + dfrac1BQ + dfrac1CR right) - dfrac3min AP, BQ, CR = dfrac6r,\ and not left( dfrac1AP + dfrac1BQ + dfrac1CR right) - dfrac3min AP, BQ, CR = dfrac6r, $$
          This is your answer



          I didn't write the answer cause it would take much time , sorry for that.






          share|cite|improve this answer




















          • Can you please provide the full solution as google books is not allowing me to view the complete solution
            – saisanjeev
            Sep 10 at 13:28










          • I just searched for it and posted the link , ok then please wait and let me try the solution myself . I am just a 12 th class student.
            – Subhajit Halder
            Sep 10 at 13:32














          up vote
          0
          down vote













          The question you wrote is wrong, it should be $$5left( dfrac1AP + dfrac1BQ + dfrac1CR right) - dfrac3min AP, BQ, CR = dfrac6r,\ and not left( dfrac1AP + dfrac1BQ + dfrac1CR right) - dfrac3min AP, BQ, CR = dfrac6r, $$
          This is your answer



          I didn't write the answer cause it would take much time , sorry for that.






          share|cite|improve this answer




















          • Can you please provide the full solution as google books is not allowing me to view the complete solution
            – saisanjeev
            Sep 10 at 13:28










          • I just searched for it and posted the link , ok then please wait and let me try the solution myself . I am just a 12 th class student.
            – Subhajit Halder
            Sep 10 at 13:32












          up vote
          0
          down vote










          up vote
          0
          down vote









          The question you wrote is wrong, it should be $$5left( dfrac1AP + dfrac1BQ + dfrac1CR right) - dfrac3min AP, BQ, CR = dfrac6r,\ and not left( dfrac1AP + dfrac1BQ + dfrac1CR right) - dfrac3min AP, BQ, CR = dfrac6r, $$
          This is your answer



          I didn't write the answer cause it would take much time , sorry for that.






          share|cite|improve this answer












          The question you wrote is wrong, it should be $$5left( dfrac1AP + dfrac1BQ + dfrac1CR right) - dfrac3min AP, BQ, CR = dfrac6r,\ and not left( dfrac1AP + dfrac1BQ + dfrac1CR right) - dfrac3min AP, BQ, CR = dfrac6r, $$
          This is your answer



          I didn't write the answer cause it would take much time , sorry for that.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Sep 7 at 9:22









          Subhajit Halder

          1039




          1039











          • Can you please provide the full solution as google books is not allowing me to view the complete solution
            – saisanjeev
            Sep 10 at 13:28










          • I just searched for it and posted the link , ok then please wait and let me try the solution myself . I am just a 12 th class student.
            – Subhajit Halder
            Sep 10 at 13:32
















          • Can you please provide the full solution as google books is not allowing me to view the complete solution
            – saisanjeev
            Sep 10 at 13:28










          • I just searched for it and posted the link , ok then please wait and let me try the solution myself . I am just a 12 th class student.
            – Subhajit Halder
            Sep 10 at 13:32















          Can you please provide the full solution as google books is not allowing me to view the complete solution
          – saisanjeev
          Sep 10 at 13:28




          Can you please provide the full solution as google books is not allowing me to view the complete solution
          – saisanjeev
          Sep 10 at 13:28












          I just searched for it and posted the link , ok then please wait and let me try the solution myself . I am just a 12 th class student.
          – Subhajit Halder
          Sep 10 at 13:32




          I just searched for it and posted the link , ok then please wait and let me try the solution myself . I am just a 12 th class student.
          – Subhajit Halder
          Sep 10 at 13:32

















           

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