$\alpha$ pythagorean theorem for $\alpha$ divergence in the probability simplex $S_n$

Question

I'm trying to understand "Methods of information geometry" of Amari and Nagaoka, p72. Consider the probability simplex $$S_n:=\{p=(p_1,\cdots,p_{n+1})\in \mathbb{R}^{n+1}~|~\sum_i^{n+1} p_i=1,~p_i\geq0\, i=0,\cdots,n+1\}$$
endowed with a $\alpha$ statistical manifold structure $(S_n, g, \nabla^{\alpha}, \nabla^{-\alpha})$ where $g$ is the fisher metric, and consider the $\alpha$ divergence $$ D_{\alpha}(p,q)=\frac{4}{1-\alpha^2}\Big[1-\sum_i p_i^{\frac{1-\alpha}{2}}q_i^{\frac{1-\alpha}{2}} \Big]$$ as well as the $\alpha$ ball $$B_\alpha(c,r):=\{p\in S_n~|~D_{\alpha}(c,p)=r\}$$ I am using the pythagorean relation, if $p$, $s\in B_\alpha(c,r)$ $$D_{\alpha}(c,s)=D_{\alpha}(c,p)+D_{\alpha}(p,s)-\frac{2}{1-\alpha^2}D_{\alpha}(c,p)D_{\alpha}(p,s)$$ with a $\alpha$ geodesic connecting $c$ and $p$ and a $-\alpha$ geodesic connecting $p$ an $s$ (lying on $B_{\alpha}$) intersecting each other orthogonally at $p$. Since $D_{\alpha}(c,s)=D_{\alpha}(c,p)=r$ this yields to the non sense formula $$D_{\alpha}(c,p)=\frac{2}{1-\alpha^2}$$ Where did I commit a mistake ?