Let $\mathfrak{g}$ be an affine lie algebra with a canonical center element $c$. I.e. if the cartan matrix is $A$ and $c_i$ are positive without a common factor such that $A (c_0,\ldots,c_n)=0$ then $c=\sum_{i=0}^n c_i h_i$.
Express the highest root $\theta$ as $\sum_{i=0}^n a_i \alpha_i$ where $\alpha_i$ are the simple roots. I am having trouble with a step on page 44 of these notes
that
the coroot corresponding to $\theta$ is $h_\theta=c-h_0$, i.e. that
$$2\frac{(\alpha, \theta)}{(\theta,\theta)}= \sum_{i >0} c_i \alpha(h_i).$$
Here $(,)$ denotes an invariant inner product normalized so that the inner product of $\theta$ with itself is $2$.
My trouble is that I don't see the relation between the canonical central element and the highest root. I feel like I am not using all the information that is available and I would appreciate if you could tell me what I am missing.