Let $a$, $b$, and $c$ be positive real numbers. Prove that

$\sqrt{a^2 - ab + b^2} + \sqrt{a^2 - ac + c^2} \ge \sqrt{b^2 + bc + c^2}.$

Under what conditions does equality occur? That is, for what values of $a$, $b$, and $c$ are the two sides equal?

Question

Let $a$, $b$, and $c$ be positive real numbers. Prove that

$\sqrt{a^2 - ab + b^2} + \sqrt{a^2 - ac + c^2} \ge \sqrt{b^2 + bc + c^2}.$

Under what conditions does equality occur? That is, for what values of $a$, $b$, and $c$ are the two sides equal?

triciaal · Answer

Did you try squaring each side  and  simplifying?

triciaal · Answer

a^2 + b^2 = (a + b)^2 - 2ab