Except for convex polygons, every simple polygon has at least one mouth.
One-Mouth Theorem
See also
Mouth, Principal Vertex, Two-Ears TheoremExplore with Wolfram|Alpha
References
Toussaint, G. "Anthropomorphic Polygons." Amer. Math. Monthly 122, 31-35, 1991.Referenced on Wolfram|Alpha
One-Mouth TheoremCite this as:
Weisstein, Eric W. "One-Mouth Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/One-MouthTheorem.html