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Polygon

Polygon is a closed plane figure which sides , , ..., , called also edges are segments. The word polygon has origin in Greek words poly, meaning "many," and gonia, meaning "angle." The points at which two polygon edges of a polygon meet are called polygon vertices. A polygon with vertices is also called an -gon.

The angle between two adjacent edges is called internal, (say ).  The sum of interior angles in an -gon is .

An exterior angle (as ) of a polygon is the angle formed externally between two adjacent edges. The sum of the external and internal angle adjacent to a vertex is clearly .

Exterior angles should not be confused with the angles  formed between an edge of a polygon and the extension of the adjacent edge (as ). Actually there two such angles at each vertex. Since both these angles are opposite, they are equal. A simple calculation shows that the sum of these angles is equal to .

A diagonal is a segment connecting two non-adjacent vertices.

A polygon is said to be simple (or Jordan) if it is described by a non-intersecting boundary, that is the only points of the plane belonging to two polygon edges of are its polygon vertices. Otherwise it is called complex For simple polygons we can define in an straightforward manner their interior and exterior.

A simple polygon is called convex if it has no internal angles greater than 180°; otherwise it is called concave or non-convex.

A polygon is called equilateral if all its edges are of the same length. A polygon with more than 4 edges can be equilateral without being convex or even simple. A polygon whose internal angles are equal is called equiangular. A polygon is called regular if all its edges and angles are equivalent.  All regular polygons with the same number of edges are similar to each other.  The measure of any interior angle of a regular n-gon is .  

A polygon vertex   of a simple polygon   is called a principal polygon vertex if the diagonal    intersects the boundary of   only at and  .

A convex polygon is called concyclic or a cyclic polygon if all the vertices lie on a single circle. A regular polygon is concyclic.

The area of a simple polygon with vertices  ,  , ..., in cartesian coordinates is given by the formula  (Meister in 1769 and Gauß in 1795)

provided the vertices are listed in order as the area is circulated in counter-clockwise fashion.  To see this divide the polygon into triangles.

If the vertices of a polygon  lattice points of a plane parallelogrammic lattice, then Pick’s theorem gives a formula for the polygon’s area based on the numbers of interior and boundary lattice points.

Lowry-Wallace-Bolyai-Gerwien theorem: Any two simple polygons of equal area can be dissected into a finite number of congruent polygonal pieces.

Note that the result implies that given two simple polygons of equal area, one can cut one of them  into finitely many polygonal pieces and rearrange (i.e. to translate or rotate) the pieces to obtain the second polygon.

Note also that analogous statement about polyhedra in three dimensions  was proven to be false by Max Dehn shortly after the Second International Congress held in Paris on August 8, 1900 where Hilbert presented it  as one (the so called Hilbert’s third problem) of  his famous collection of 23 problems.

The three dimensional analog of a polygon is called polyhedron, in four dimensions it is called a polychoron, and generally in a higher  dimension it is called an -dimensional  polytope.

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