It's not clear to me why a geometrical proof is so hard to find. The definition says a polygon is convex if we can connect any pair of two points of the polygon with a line that's contained in that polygon. In most sources, for unknown reason, it is treated as an obvious fact.
This is the proof I've found – page 2. Could you verify whether it is correct? I have doubts with regard to this proof – there's a contradiction if we allow angles $> 180$, but how do we know if we can connect two points inside the polygon with a line inside the polygon, if it has angles $<180$? Maybe we would arrive at contradiction if we assumed angles $>170$. We should prove that all angles $< 180$ are okay here.
Best Answer
Let $A$ be any vertex of the polygon. Let $B$, $C$ be the vertices adjacent to $A$.
Angle $BAC$ (the internal angle) is inside the non-degenerate triangle $BAC$, and hence is less than $180^{\circ}$.