A Graph is $2$-connected iff each two vertices lie on the same cycle.
Proof: left to right: Graph is $2$-connected if each $2$ vertices are connected with at least two disjoint paths(follows from Menger's Theorem). Take the vertices $u$ and $v$ and let this two paths be $w_1=(u,u_1,...,u_n,v)$ and $w_2=(u,w_1,...,w_n,v)$ where only $u$ and $v$ are vertices in common. $w_2=(u,w_1,...w_n,v)$ is equivalent to $w_2=(v,w_n,...,w_1,u)$. We can bring them together and get $w_1=(u,u_1,...,v,w_n,...,w_1,u)$, our cycle. right to left: We have the cycle $w_1=(u,u_1,...,v,w_n,...,w_1,u)$ where both our vertices lie. Very similar to what we did above but now reversed we divide it and get our two disjoint paths.
Would this proof be correct?
Thanks in advance