For any triangle ABC with sides a,b,c prove that a cos ⁡ A + b...

Jovany Clayton

Jovany Clayton

Answered

2022-06-30

For any triangle ABC with sides a,b,c prove that
a cos A + b cos B + c cos C a + b + c 2
How to initiate this question?

Answer & Explanation

kawiarkahh

kawiarkahh

Expert

2022-07-01Added 15 answers

If a b c then α β γ and since cos is a decreasing function on [ 0 , π ] , we obtain:
cos α cos β cos γ .
Thus, since
cos α + cos β + cos γ = 1 + r R 3 2 ,
by Chebyshov we obtain:
c y c a cos α 1 3 ( a + b + c ) ( cos α + cos β + cos γ ) a + b + c 2 .
Done!
Esmeralda Lane

Esmeralda Lane

Expert

2022-07-02Added 7 answers

We need to prove that
c y c a ( b 2 + c 2 a 2 ) 2 b c a + b + c 2
or
c y c a 2 ( b 2 + c 2 a 2 ) c y c a 2 b c
or
c y c ( a 4 2 a 2 b 2 + a 2 b c ) 0
or
c y c ( a 4 a 3 b a 3 c + a 2 b c ) + c y c ( a 3 b + a 3 c 2 a 2 b 2 ) 0 ,
which is Schur and c y c a b ( a b ) 2 0.
Done!

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