Editing Midpoint (section)

===General polygons===

*A [[regular polygon]] has an [[inscribed circle]] which is [[tangent]] to each side of the polygon at its midpoint.

*In a regular polygon with an even number of sides, the midpoint of a [[diagonal]] between opposite vertices is the polygon's center.

*The [[midpoint-stretching polygon]] of a [[cyclic polygon]] {{mvar|P}} (a [[polygon]] whose vertices all fall on the same circle) is another cyclic polygon inscribed in the same circle, the polygon whose vertices are the midpoints of the [[circular arc]]s between the vertices of {{mvar|P}}.<ref name="dhz">{{Citation |last1=Ding |first1=Jiu |last2=Hitt |first2=L. Richard |last3=Zhang |first3=Xin-Min |date=1 July 2003 |title=Markov chains and dynamic geometry of polygons |journal=Linear Algebra and Its Applications |volume=367 |pages=255–270 |doi=10.1016/S0024-3795(02)00634-1 |url=http://www.rhitt.com/research/markov.pdf |access-date=19 October 2011}}.</ref> Iterating the midpoint-stretching operation on an arbitrary initial polygon results in a sequence of polygons whose shapes converge to that of a [[regular polygon]].<ref name="dhz"/><ref>{{Citation |first1=Francisco |last1=Gomez-Martin |first2=Perouz |last2=Taslakian |first3=Godfried T. |last3=Toussaint|author3-link=Godfried Toussaint |year=2008 |contribution=Convergence of the shadow sequence of inscribed polygons|title=18th Fall Workshop on Computational Geometry |publisher=Artesa |isbn=978-84-8181-227-5 |url=http://oa.upm.es/4442/}}</ref>