Nesbitt's Inequality

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

Then:
 * $\dfrac {a} {b+c} + \dfrac {b} {a+c} + \dfrac {c} {a+b} \ge \dfrac 3 2$

Proof
These are the arithmetic mean and the harmonic mean of $\dfrac {1} {b+c}$, $\dfrac {1} {a+c}$ and $\dfrac {1} {a+b}$.

The arithmetic mean is never less than the harmonic mean, so the last inequality is true, thus Nesbitt's inequality holds.