Union of two circles is star domain with center any of the circles center

Question

I'm trying to prove that a power series $f(z)=\sum_{i=0}^\infty a_n (z-a)^n$ with a positive finite radius of convergence $r$, then $\exists z_0, |z_0-a|=r$ such that $\nexists \Omega, D(a,r) \subset \Omega, z_0 \in \Omega$ and $\tilde{f} \in \mathcal{H}(\Omega)$ with $\tilde{f}|_{D(a,r)}=f$. Any idea of a simple proof of the fact state in the title?