2

What is the simplest way to prove that if

$ A = \frac{4 b c-a^2}{b c+2 a^2}, B = \frac{4 a c-b^2}{a c+2 b^2}, C = \frac{4 a b-c^2}{a b+2 c^2}, a+b+c =0$

then $ A+B+C = 3 \land ABC=1$ ?

This is not a homework, I'm just trying to get better at math. My naive way to start would be to add A, B and C and try to figure something out from that but I think that there must be some more elegant way. Any hints?

1osmi
  • 377

1 Answers1

3

Hint: $4bc-a^2 = -(b-c)^2$, $bc+2a^2=(b-a)(c-a)$, $\ldots$

njguliyev
  • 14,473
  • 1
  • 26
  • 43