A maximal planar graph $G$ with at least 3 vertices is a simple finite planar graph for which we cannot add any new edge $e$ such that $G \cup e$ is still planar. Is there an easy and rigorous way to prove that such a graph is connected (between every 2 vertices there is a path)?
I know that such a graph is a plane triangulation. But I don't wish to use the fact that this means $G$ is 3-connected since it seems overkill. I also know there's an intuitive argument where I look at the "face with the disconnected components" and "add an edge within it connecting them". Is there a way to make this rigorous (using topological facts like in Diestel) or maybe a better proof?