Let $X$ a compact connected Hausdorff space, and that $D$ is a proper compact connected subset of $X$ with non-empty interior. I am trying to show that there exists a compact connected subset $C$ of $X$ such that $C \not \subseteq D$, $D \not \subseteq C$, and $C \cap D \neq \emptyset$.
In an earlier question I showed that for a proper subset $E$ of a compact connected Hausdorff space $X$, and a component $F$ of $E$, we have that $E \cap F \setminus int(D) \neq \emptyset$. Perhaps this is helpful? So far I have not gotten anywhere.