A geometric inequality to triangle $ABC$

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If $a, b, c$ be the sides of a triangle $ABC$, I have to prove that



$sumlimits_cycl^ (frac ab+c)^2 + frac 3 A^2abcs ge frac 98$



where $A,s$ the area and the semiperemeter. I tried to use the Ravi's substitution, but the inequality was then quite hard to solve. It might be a differente method. Thank you




inequality contest-math




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  • 1




    Use the $uvw$ method.
    – Michael Rozenberg
    Jul 19 at 14:03






  • 3




    Isn't this inequality equivalent to the one you asked in another question?
    – achille hui
    Jul 19 at 14:11






  • 1




    Yes of course, but he wants another method. See please better a starting post.
    – Michael Rozenberg
    Jul 19 at 14:16







  • 1




    @Steven I found a proof by SOS, but it's very ugly.
    – Michael Rozenberg
    Jul 19 at 15:57










  • Thank you @MichaelRozenberg, you are very strong
    – Steven
    Jul 19 at 20:44














up vote
0
down vote

favorite












If $a, b, c$ be the sides of a triangle $ABC$, I have to prove that



$sumlimits_cycl^ (frac ab+c)^2 + frac 3 A^2abcs ge frac 98$



where $A,s$ the area and the semiperemeter. I tried to use the Ravi's substitution, but the inequality was then quite hard to solve. It might be a differente method. Thank you




inequality contest-math




share|cite|improve this question

















  • 1




    Use the $uvw$ method.
    – Michael Rozenberg
    Jul 19 at 14:03






  • 3




    Isn't this inequality equivalent to the one you asked in another question?
    – achille hui
    Jul 19 at 14:11






  • 1




    Yes of course, but he wants another method. See please better a starting post.
    – Michael Rozenberg
    Jul 19 at 14:16







  • 1




    @Steven I found a proof by SOS, but it's very ugly.
    – Michael Rozenberg
    Jul 19 at 15:57










  • Thank you @MichaelRozenberg, you are very strong
    – Steven
    Jul 19 at 20:44












up vote
0
down vote

favorite









up vote
0
down vote

favorite











If $a, b, c$ be the sides of a triangle $ABC$, I have to prove that



$sumlimits_cycl^ (frac ab+c)^2 + frac 3 A^2abcs ge frac 98$



where $A,s$ the area and the semiperemeter. I tried to use the Ravi's substitution, but the inequality was then quite hard to solve. It might be a differente method. Thank you




inequality contest-math




share|cite|improve this question













If $a, b, c$ be the sides of a triangle $ABC$, I have to prove that



$sumlimits_cycl^ (frac ab+c)^2 + frac 3 A^2abcs ge frac 98$



where $A,s$ the area and the semiperemeter. I tried to use the Ravi's substitution, but the inequality was then quite hard to solve. It might be a differente method. Thank you





inequality contest-math





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited 19 hours ago









amWhy

189k25219431




189k25219431









asked Jul 19 at 13:33









Steven

41019




41019







  • 1




    Use the $uvw$ method.
    – Michael Rozenberg
    Jul 19 at 14:03






  • 3




    Isn't this inequality equivalent to the one you asked in another question?
    – achille hui
    Jul 19 at 14:11






  • 1




    Yes of course, but he wants another method. See please better a starting post.
    – Michael Rozenberg
    Jul 19 at 14:16







  • 1




    @Steven I found a proof by SOS, but it's very ugly.
    – Michael Rozenberg
    Jul 19 at 15:57










  • Thank you @MichaelRozenberg, you are very strong
    – Steven
    Jul 19 at 20:44












  • 1




    Use the $uvw$ method.
    – Michael Rozenberg
    Jul 19 at 14:03






  • 3




    Isn't this inequality equivalent to the one you asked in another question?
    – achille hui
    Jul 19 at 14:11






  • 1




    Yes of course, but he wants another method. See please better a starting post.
    – Michael Rozenberg
    Jul 19 at 14:16







  • 1




    @Steven I found a proof by SOS, but it's very ugly.
    – Michael Rozenberg
    Jul 19 at 15:57










  • Thank you @MichaelRozenberg, you are very strong
    – Steven
    Jul 19 at 20:44







1




1




Use the $uvw$ method.
– Michael Rozenberg
Jul 19 at 14:03




Use the $uvw$ method.
– Michael Rozenberg
Jul 19 at 14:03




3




3




Isn't this inequality equivalent to the one you asked in another question?
– achille hui
Jul 19 at 14:11




Isn't this inequality equivalent to the one you asked in another question?
– achille hui
Jul 19 at 14:11




1




1




Yes of course, but he wants another method. See please better a starting post.
– Michael Rozenberg
Jul 19 at 14:16





Yes of course, but he wants another method. See please better a starting post.
– Michael Rozenberg
Jul 19 at 14:16





1




1




@Steven I found a proof by SOS, but it's very ugly.
– Michael Rozenberg
Jul 19 at 15:57




@Steven I found a proof by SOS, but it's very ugly.
– Michael Rozenberg
Jul 19 at 15:57












Thank you @MichaelRozenberg, you are very strong
– Steven
Jul 19 at 20:44




Thank you @MichaelRozenberg, you are very strong
– Steven
Jul 19 at 20:44















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