Surprising but easy

If

$$\frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b} = 1 $$

Show that

$$\frac{a^2}{b+c} + \frac{b^2}{c+a} + \frac{c^2}{a+b} = 0 $$

Solution [Click here to expand/collapse]

Consider $$a + b + c = (a+b+c)\left(\frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b}\right)$$
This is the same as $$ a + b + c + \frac{a^2}{b+c} + \frac{b^2}{c+a} + \frac{c^2}{a+b}$$
And thus $$ \frac{a^2}{b+c} + \frac{b^2}{c+a} + \frac{c^2}{a+b} = 0$$