Making planar graph biconnected
One way to augmenting an embedded planar graph (i.e. a plane graph) to become biconnected, while preserving the embedding, is

for each articulation vertex v:
  for each two edges vu and vw that are consecutive in the cyclic order around v:
    if u and w belong to different biconnected components of G:
      add edge uw

You have to be a little careful about the ordering of operations, though, because if you did this simultaneously at both endpoints of a bridge edge you could create a non-planarity. You have to do them one at a time.

This method can add significantly fewer edges than triangulation: each new edge reduces the number of biconnected components by one, so the total number of edges added is less than the original number of biconnected components.