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For any convex polygon, all the diagonals are inside the polygon, but for re-entrant polygons, some diagonals are outside of the polygon. Any n -sided polygon ( n ≥ 3), convex or concave , has n ( n − 3 ) 2 {\displaystyle {\tfrac {n(n-3)}{2}}} total diagonals, as each vertex has diagonals to all other vertices except itself and the two ...
The polygon is the convex hull of its edges. Additional properties of convex polygons include: The intersection of two convex polygons is a convex polygon. A convex polygon may be triangulated in linear time through a fan triangulation, consisting in adding diagonals from one vertex to all other vertices.
[2] [3] A kite may also be called a dart, [4] particularly if it is not convex. [5] [6] Every kite is an orthodiagonal quadrilateral (its diagonals are at right angles) and, when convex, a tangential quadrilateral (its sides are tangent to an inscribed circle). The convex kites are exactly the quadrilaterals that are both orthodiagonal and ...
A non-convex regular polygon is a regular star polygon. The most common example is the pentagram , which has the same vertices as a pentagon , but connects alternating vertices. For an n -sided star polygon, the Schläfli symbol is modified to indicate the density or "starriness" m of the polygon, as { n / m }.
Other computational problems studied for simple polygons include constructions of the longest diagonal or the longest line segment interior to a polygon, [13] of the convex skull (the largest convex polygon within the given simple polygon), [29] [30] and of various one-dimensional skeletons approximating its shape, including the medial axis [31 ...
Convex quadrilaterals by symmetry, represented with a Hasse diagram. In a convex quadrilateral all interior angles are less than 180°, and the two diagonals both lie inside the quadrilateral. Irregular quadrilateral (British English) or trapezium (North American English): no sides are parallel. (In British English, this was once called a trapezoid
A principal diagonal of a hexagon is a diagonal which divides the hexagon into quadrilaterals. In any convex equilateral hexagon (one with all sides equal) with common side a, there exists [11]: p.184, #286.3 a principal diagonal d 1 such that and a principal diagonal d 2 such that
Euler's quadrilateral theorem or Euler's law on quadrilaterals, named after Leonhard Euler (1707–1783), describes a relation between the sides of a convex quadrilateral and its diagonals. It is a generalisation of the parallelogram law which in turn can be seen as generalisation of the Pythagorean theorem.