Prove of triangulation.

Question

I am exploring the proof by induction that any polygon can be triangulated.

There is a place in the proof when we take the leftmost vertex v and two its neighbors u and w.

We connect u and w and it turns out that the edge uw is not lying in the polygon. Then we take the vertex which is farthest from uw. And here is my question is it okay to take not the farthest one, but the leftmost one?

It seems to be an easy to answer question if you have a counter example, but still I can not find one.

Answer 1

Here is a counterexample from the paper Efficient Triangulation of Simple Polygons by Godfried Toussaint: $w$ is the leftmost vertex inside the triangle $x_{i-1} x_i x_{i+1}$ but $w$ is not visible from $x_i$: