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 variables a, b, and c, are positive real numbers. Prove that:

 

sqrt(a^2-ab+b^2) + sqrt(a^2-ac+c^2) is greater than or equal to sqrt(b^2+bc+c^2)

 

Under what conditions does equality occur? That is, for what values are both sides of the inequality equal?

 

I would really appreciate it if I could get an explanation and not just an answer, thank you!

 May 5, 2021
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By the Cosine Law, a^2 - ab + b^2 = c^2 \cos C.  Plug in similar expressions, and the rest is easy.

 May 5, 2021

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