Let $G$ be a 2-connected, simple, plane graph with $\delta(G)>2$. Either prove the dual $G^*$ is simple or find a counterexample.
I am stuck on this problem, I just something to start with and then hopefully I should figure it out. I tried to draw such a graph many times and I think that this kind of graph does exist. So, there is no counterexample for it. Now, all I need is something to start my proof.
Thank you for your help.
Here is a counterexample:
This is a 2-connected simple planar graph with minimum degree $3$, but the length-$6$ internal face has two borders with the external face, so in the dual there will be two edges between the corresponding vertices.
What you can say about 2-connected planar graphs is that the dual will not have loops. This would correspond to an edge in the planar graph with the same face on both sides, so deleting that edge would disconnect its endpoints. So the graph would not even be 2-edge-connected, much less 2-connected.