Why simple polygons in plane have this property?

Question

If we are given a simple polygon $P$ in the plane by the points $A_1, A_2, \dots, A_n$. How can we prove that there are $3$ consecutive points $A_i, A_{i+1}, A_{i+2}$ (if $i = n$, for $A_{i + 1}$ and $A_{i+2}$ I mean $A_1$ and $A_2$ of course) such that there isn't another point $A_j$ in the triangle $A_i A_{i+1}A_{i+2}$?

Answer 1

Imagine a simple polygon with at least one exception:

It is easy to prove that this is impossible with 4 or less vertices, so remove any offending vertex and replace with a line connecting the other two. This reduces to one of the cases mentioned before, and was part of the original polygon.