Seems to me, at least the way you've phrased it, is just an extension of the idea that the shortest distance between two points is a straight line. If you have an obstructing polygon that you are not allowed to bisect, you are now looking at making the shortest distance between your original starting point and the polygon, then the shortest distance along the polygon, then the shortest distance between that point and your endpoint. Since the line segments between vertices represent the shortest path between the points on the interfering polygon, traveling between those would represent the shortest path around the obstruction.
You are basically describing my thought process every time I walk to/from work on a miserable day (cold, hot, rainy, etc).
1) The shortest distance is a straight line from where you are at to the end point.
2) If that path is blocked, then you have to go around the obstacle.
3) It should be obvious that the shortest distance is NOT a wide path around the obstacle but instead a path that just grazes the outside of that obstacle.
4) If I were in fact infinitely narrow, the shortest walking path would put me exactly on at least one corner of that obstacle (a vertex of the polygon).
5) Go back to #1.
I can't count the number of times I've thought this out while walking, hoping to come up with a shorter path. It inevitably leads to me thinking about building/programing a robot to run a maze.