Does a concave polygon always include a convex poligon?

Question

Does a concave n-gon, for $n\geq 3$ always include a convex polygon? Rather simplified, a triangle? I have defined break as breaking a convex polygon to a concave one from numerous points, like take a regular pentagon and break it from 2 points. I have taken a convex n-gon and observed that it includes $n-2$ triangles, I have also observed that I can break it maximally from $\left \lfloor \frac{n}{2}\right \rfloor$ points. And that the number of triangles it includes don't change. Thus yes. Although I don't know how to prove my observations and whether I can find a counter example or sth. Any suggestions?