Prove that an edge in planar graph cannot belong to three faces using the Jordan curve theorem

Question

Given a planar drawing $D$ of a graph $G$, how can I prove that an edge cannot belong to three faces in the drawing $D$? I am looking for a proof that

1. uses the Jordan curve theorem, and

2. does not assume $D$ is a straight line drawing of $G$.

This question was asked here, and this answer suggests that the Jordan curve theorem can be used to give a proof. Another answer uses the Jordan curve theorem, but assumes that the edges of $G$ are drawn as line segments in $D$.