Consider a polygon $A_{0} \dots A_{n}$. Why does there exist diagonal $A_{i}A_{j}$ without intersection of polygon. It's looks obvious but why?
A convex polygon is a non-empty intersection of convex sets (half-planes), hence it is convex. In particular, if we consider the segment joining $A_i$ and $A_j$ with $|i-j|>1$, such segment lies inside the polygon, since it is the convex envelope of $A_i$ and $A_j$.