How do I show that cos 2 ⁡ A cos 2 ⁡ B + cos 2 ⁡ B cos 2 ⁡ C + cos 2 ⁡ C cos 2 ⁡ A ≥ 4 ( cos 2 ⁡ A + cos 2 ⁡ B + cos 2 ⁡ C )? A , B , C be the angles of an acute triangle.How should I approach this kind of "geometric inequalities"? I've considered substituting the cosines using the law of the cosines and then use the Ravi transformation to turn it into an algebraic one, but that seems too tedious and unlikely to yield any beautiful solution. Any hints will be appreciated!

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How do I show that                     cos        2            ⁡      A                      cos        2            ⁡      B        +                    cos        2            ⁡      B                      cos        2            ⁡      C        +                    cos        2            ⁡      C                      cos        2            ⁡      A        ≥  4  (      cos    2    ⁡  A  +      cos    2    ⁡  B  +      cos    2    ⁡  C  )?  A  ,  B  ,  C be the angles of an acute triangle.How should I approach this kind of &quot;geometric inequalities&quot;? I&#039;ve considered substituting the cosines using the law of the cosines and then use the Ravi transformation to turn it into an algebraic one, but that seems too tedious and unlikely to yield any beautiful solution. Any hints will be appreciated!

noruegezajl00y · Accepted Answer

By AM-GM easy to show that       x    y    +      y    z    +      z    x    ≥            x      +      y      +      z                      x        y        z            3       for all positives   x,   y and   z and  cos  ⁡  α  cos  ⁡  β  cos  ⁡  γ  ≤      1    8  Hence,       ∑          c      y      c                          cos        2            ⁡      α                      cos        2            ⁡      β        ≥                    cos        2            ⁡      α      +              cos        2            ⁡      β      +              cos        2            ⁡      γ                                cos          2                ⁡        α                  cos          2                ⁡        β                  cos          2                ⁡        γ            3        ≥  4      (          cos      2        ⁡    α    +          cos      2        ⁡    β    +          cos      2        ⁡    γ    )  . Done!

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