I'm trying to prove the following:
Let $G$ be a connected graph of order at least 3 with no cut-vertices. Then, every pair of edges in $G$ lies on a common cycle.
My starting point is to use the fact that every pair of vertices of $G$ lies on a common cycle. This tells us that if $e = wx, f = yz$ are two edges in $G$, then, every pair of $w,x,y,z$ lies on a common cycle, but I'm getting stuck on showing that $e$ and $f$ are on a common cycle.
Any hints or tips?