Two ears theorem

It is frequently attributed to Gary H. Meisters, but was proved earlier by Max Dehn.

A simple polygon is a simple closed curve in the Euclidean plane consisting of finitely many line segments in a cyclic sequence, with each two consecutive line segments meeting at a common endpoint, and no other intersections.

By the Jordan curve theorem, it separates the plane into two regions, one of which (the interior of the polygon) is bounded.

An ear of a polygon is defined as a triangle formed by three consecutive vertices

Conversely, if a polygon is triangulated, the weak dual of the triangulation (a graph with one vertex per triangle and one edge per pair of adjacent triangles) will be a tree and each leaf of the tree will form an ear.

Since every tree with more than one vertex has at least two leaves, every triangulated polygon with more than one triangle has at least two ears.

Thus, the two ears theorem is equivalent to the fact that every simple polygon has a triangulation.

By testing all neighbors of all vertices, it is possible to find all the ears of a triangulated simple polygon in linear time.

[5] An ear is called exposed when its central vertex belongs to the convex hull of the polygon.

A principal vertex for which this line segment lies outside the polygon is called a mouth.

Analogously to the two ears theorem, every non-convex simple polygon has at least one mouth.