I have to proof the following
Let $v_1,v_2,v_3$ be any 3 vertices on a 3-vertex-connected graph $G=(V,E)$. Then they belong to a circle $C$
What I got so far.
Any two vertices $ v_1,v_2 \in V$ are on a circle $C$. There exist 3 disjoint vertices $u_1, u_2,u_3$ in $C$ such that paths from $u$ to $u_i$ with $u \notin C$ are pairwise disjoint and $u_i$ is the only vertex in that path belonging to $C$ .
How can I continue from here ?