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

Misael Matthews 2022-06-25 Answered
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
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Answers (2)

Nola Rivera
Answered 2022-06-26 Author has 21 answers
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's inequality.
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Damon Stokes
Answered 2022-06-27 Author has 6 answers
Since a b + a c + b c c y c ( 2 a b a 2 ) it'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 ) > 0 ,
it's enough to prove that
c y c ( 2 a b a 2 ) 4 3 S
or
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!
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