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 $\dfrac 1 {a + b}$.

From Arithmetic Mean Never Less than Harmonic Mean the last inequality is true.

Thus Nesbitt's inequality holds.