I am stuck in the following problem from Humphreys. Let $\alpha, \beta$ be roots in a root system $\Phi$. Let the $\alpha$-string through $\beta$ be $\beta - r\alpha, \ldots, \beta + q\alpha$ and let the $\beta$-string through $\alpha$ be $\alpha - r'\beta, \ldots, \alpha + q'\beta$. Prove that \begin{equation}\frac{q(r+1)}{(\beta,\beta)} = \frac{q'(r'+1)}{(\alpha,\alpha)},\end{equation} where $(-,-)$ denotes the inner product on the ambient space.
Follow the discussion in Humphreys on page 45, I can show that $$r-q = \frac{2(\beta,\alpha)}{(\alpha,\alpha)} \hspace{5mm} \text{and} \hspace{5mm} r' - q' = \frac{2(\alpha,\beta)}{(\beta,\beta)}.$$ This implies in particular that $$(\alpha,\alpha)(r-q) = (\beta,\beta)(r'-q').$$ However, I have been unsuccessful with obtaining the desired equality from this one, no matter what I try. Even though my tools are quite limited (i.e. I have only the definition of a root system and a few basic lemmas), I am unable to find a path to solve this problem. Any help you can give is much appreciated.
Look at the excersise 9.7 All you need to do is to check the fact for two-dimensional root systems. But you have a complete list of them. (Just noticed the date of your question. Well, maybe you are still intrested)