<s>[INST] Find the value of $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$ 
Suppose $(a, b, c)\in\Bbb R^3$, $a,b,c$ are all nonzero, and we have $
\sqrt{a+b}+\sqrt{b+c}=\sqrt{c+a}$. Find the value of $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}.$$

Here is my attempt: 
$$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{bc+ac+ab}{abc}.$$
I am having trouble in figuring out the best approach to simplify $
\sqrt{a+b}+\sqrt{b+c}=\sqrt{c+a}$
so that I can find the value of $
\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$ . Hope somebody has an idea.
 [/INST] Solving your equation $$\sqrt{a+b}+\sqrt{b+c}=\sqrt{c+a}$$ for $c$ we get
$$c=-\frac{ab}{a+b}$$ then we get
$$\frac{1}{a}+\frac{1}{b}-\frac{a+b}{ab}=0$$
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