For a, b, c the sides of a triangle, prove | a b − b...

Hayley Bernard

Hayley Bernard

Answered

2022-07-14

For a, b, c the sides of a triangle, prove | a b b a + b c c b + c a a c | < 1
I initially thought of changing it in terms of sine using the law of sines, which gives me
| sin A sin B sin B sin A + sin B sin C sin C sin B + sin C sin A sin A sin C | < 1
However, from there I'm not sure how to proceed. Any tips?

Answer & Explanation

decoratesuw

decoratesuw

Expert

2022-07-15Added 11 answers

Hint: Using the substitution a = x + y , b = y + z , c = z + x (if a , b , c are sides of a triangle then x , y , z always exist and are positive) we get
| c y c ( a b b a ) | = | ( x y ) ( y z ) ( z x ) ( x + y ) ( y + z ) ( z + x ) | < 1
yasusar0

yasusar0

Expert

2022-07-16Added 3 answers

By an inequality of the triangle | c y c ( a b a c ) | = | c y c ( a 2 c a 2 b ) | a b c = c y c | a b | a b c = c y c | a b | c < 1

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