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| [[File:Rectangle definition.svg|thumb|An equiangular quadrilateral]]
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| In [[Euclidean geometry]], an '''equiangular polygon''' is a [[polygon]] whose vertex angles are equal. If the lengths of the sides are also equal then it is a [[regular polygon]].
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| The only equiangular triangle is the [[equilateral triangle]]. [[Rectangle]]s, including the square, are the only equiangular [[quadrilateral]]s (four-sided figures).<ref name="ball">{{citation|first=Derek|last=Ball|title=Equiangular polygons|journal=The Mathematical Gazette|volume=86|issue=507|year=2002|pages=396–407|jstor=3621131}}.</ref>
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| For an equiangular ''n''-gon each angle is 180° − <math>\tfrac{1}{n}</math>(360°); this is the ''equiangular polygon theorem''.
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| [[Viviani's theorem]] holds for equiangular polygons:<ref name="ref1">[http://arxiv.org/abs/0903.0753v3 Elias Abboud "On Viviani’s Theorem and its Extensions"] pp. 2, 11</ref>
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| :''The sum of distances from an interior point to the sides of an equiangular polygon does not depend on the location of the point, and is that polygon's invariant.''
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| A rectangle (equiangular quadrilateral) with integer side lengths may be tiled by [[unit square]]s, and an equiangular [[hexagon]] with integer side lengths may be tiled by unit [[equilateral triangle]]s. Some but not all equilateral [[dodecagon]]s may be tiled by a combination of unit squares and equilateral triangles; the rest may be tiled by these two shapes together with [[rhombus|rhombi]] with 30 and 150 degree angles.<ref name="ball"/>
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| A [[cyclic polygon]] is equiangular if and only if the alternate sides are equal (that is, sides 1, 3, 5, ... are equal and sides 2, 4, ... are equal). Thus if ''n'' is odd, a cyclic polygon is equiangular if and only if it is regular.<ref>De Villiers, Michael, "Equiangular cyclic and equilateral circumscribed polygons", [[Mathematical Gazette]] 95, March 2011, 102-107.</ref>
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| ==References==
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| {{reflist}}
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| *Williams, R. ''The Geometrical Foundation of Natural Structure: A Source Book of Design''. New York: [[Dover Publications]], 1979. p. 32
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| ==External links==
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| *{{MathWorld|title=Equiangular Polygon|urlname=EquiangularPolygon}}
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| *[http://www.cut-the-knot.org/Curriculum/Geometry/EquiangularPoly.shtml A Property of Equiangular Polygons: What Is It About?] a discussion of Viviani's theorem at [[Cut-the-knot]].
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| [[Category:Polygons]]
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| {{Elementary-geometry-stub}}
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