1 a + b </mrow> </mfrac> + 1 a

Sam Hardin 2022-07-10 Answered
1 a + b + 1 a + c = 3 a + b + c in a triangle
Find the angle α of a triangle with sides a, b and c for which the equality
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Answers (2)

Valeria Wolfe
Answered 2022-07-11 Author has 11 answers
Step 1
cos α = b 2 + c 2 a 2 2 b c
after simplifying the given equality and plug in something, but this does not seem to be easy.
1 a + b + 1 a + c = 3 a + b + c 2 a + b + c ( a + b ) ( a + c ) = 3 a + b + c ( 2 a + b + c ) ( a + b + c ) = 3 ( a + b ) ( a + c )
Step 2
The last line, you get b 2 + c 2 = a 2 + b c , so the cos α = 1 / 2cosα=1/2 .

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Logan Wyatt
Answered 2022-07-12 Author has 5 answers
Step 1
Multiplying both sides of the equation by
( a + b ) ( a + c ) ( a + b + c ) , we get
( a + c ) ( a + b + c ) + ( a + b ) ( a + b + c ) = 3 ( a + c ) ( a + b ) a 2 + a b + 2 a c + b c + c 2 + a 2 + 2 a b + a c + b 2 + b c = 3 a 2 + 3 a b + 3 a c + 3 b c 2 a 2 + 3 a b + 3 a c + b 2 + 2 b c + c 2 = 3 a 2 + 3 a b + 3 a c + 3 b c b c = a 2 + b 2 + c 2
Hence, we have cos α = 1 2 . Since we know 0 < α < π, we have α = π 3

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