Prove this inequality where a, b and c are sides of a triangle and S its Area. a b + b c + c a 4 S ≥ ctg ⁡ π 6

Question

Prove this inequality where a, b and c are sides of a triangle and S its Area.            a      b      +      b      c      +      c      a              4      S        ≥  ctg  ⁡      π    6

Nola Rivera · Accepted Answer

By the sine theorem, the given inequality is equivalent to                    (1)                              1                      sin            ⁡            A                          +                  1                      sin            ⁡            B                          +                  1                      sin            ⁡            C                          ≥        2                  3                    and since       1          sin      ⁡      (      x      )       is a convex function on the interval   (  0  ,  π  ), (1) is a straightforward consequence of Jensen&#039;s inequality.

Damon Stokes · Accepted Answer

Since   a  b  +  a  c  +  b  c  ≥      ∑          c      y      c        (  2  a  b  −      a    2    ) it&#039;s just       ∑          c      y      c        (  a  −  b      )    2    ≥  0  and      ∑          c      y      c        (  2  a  b  −      a    2    )  =      ∑          c      y      c        a  (  b  +  c  −  a  )  &amp;gt;  0  ,it&#039;s enough to prove that      ∑          c      y      c        (  2  a  b  −      a    2    )  ≥  4      3    Sor      ∑          c      y      c        (  2  a  b  −      a    2    )  ≥      3          ∑              c        y        c              (    2          a      2              b      2        −          a      4        )  or      ∑          c      y      c        (      a    4    −      a    3    b  −      a    3    c  +      a    2    b  c  )  ≥  0  ,which is Schur.Done!

Prove this inequality where a, b and c are sides of a triangle and S its Area. a

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