Editing Simple polygon (section)

==Properties==
The ''[[internal angle]]'' of a simple polygon, at one of its vertices, is the angle spanned by the interior of the polygon at that vertex. A vertex is ''convex'' if its internal angle is less than <math>\pi</math> (a straight angle, 180°) and ''concave'' if the internal angle is greater than <math>\pi</math>. If the internal angle is <math>\theta</math>, the ''[[external angle]]'' at the same vertex is defined to be its [[supplementary angle|supplement]] <math>\pi-\theta</math>, the turning angle from one directed side to the next. The external angle is positive at a convex vertex or negative at a concave vertex. For every simple polygon, the sum of the external angles is <math>2\pi</math> (one full turn, 360°). Thus the sum of the internal angles, for a simple polygon with <math>n</math> sides is <math>(n-2)\pi</math>.{{r|rich2}}

[[File:Triangulation 3-coloring.svg|thumb|left|A triangulated polygon with 11 vertices: 11 sides and 8 diagonals form 9 triangles.]]
Every simple polygon can be partitioned into non-overlapping triangles by a subset of its diagonals. When the polygon has <math>n</math> sides, this produces <math>n-2</math> triangles, separated by <math>n-3</math> diagonals. The resulting partition is called a ''[[polygon triangulation]]''.{{r|meisters}} The shape of a triangulated simple polygon can be uniquely determined by the internal angles of the polygon and by the [[cross-ratio]]s of the [[quadrilateral]]s formed by pairs of triangles that share a diagonal.{{r|cross-ratio}}

According to the [[two ears theorem]], every simple polygon that is not a triangle has at least two ''ears'', vertices whose two neighbors are the endpoints of a diagonal.{{r|meisters}} A related theorem states that every simple polygon that is not a [[convex polygon]] has a ''mouth'', a vertex whose two neighbors are the endpoints of a line segment that is otherwise entirely exterior to the polygon. The polygons that have exactly two ears and one mouth are called ''[[anthropomorphic polygon]]s''.{{r|toussaint}}

[[File:Art gallery problem.svg|thumb|This 42-vertex polygonal art gallery is entirely visible from cameras placed at the 4 marked vertices.]]
According to the [[art gallery theorem]], in a simple polygon with <math>n</math> vertices, it is always possible to find a subset of at most <math>\lfloor n/3\rfloor</math> of the vertices  with the property that every point in the polygon is visible from one of the selected vertices. This means that, for each point <math>p</math> in the polygon, there exists a line segment connecting <math>p</math> to a selected vertex, passing only through interior points of the polygon. One way to prove this is to use [[graph coloring]] on a triangulation of the polygon: it is always possible to color the vertices with three colors, so that each side or diagonal in the triangulation has two endpoints of different colors. Each point of the polygon is visible to a vertex of each color, for instance one of the three vertices of the triangle containing that point in the chosen triangulation. One of the colors is used by at most <math>\lfloor n/3\rfloor</math> of the vertices, proving the theorem.{{r|fisk}}