Equilateral polygon

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In geometry, an equilateral polygon is a polygon which has all sides of the same length.

For instance, an equilateral triangle is a triangle of equal edge lengths. All equilateral triangles are similar to each other, and have 60 degree internal angles.

Equilateral Triangle.svg

An equilateral quadrilateral is a rhombus, which includes the square as a special case.

Rhombus.svg

An equilateral polygon that is also equiangular is a regular polygon.

An equilateral polygon which is cyclic (its vertices are on a circle) is a regular polygon (a polygon that is both equilateral and equiangular).

A tangential polygon (one that has an incircle tangent to all its sides) is equilateral if and only if the alternate angles are equal (that is, angles 1, 3, 5, ... are equal and angles 2, 4, ... are equal). Thus if the number of sides n is odd, a tangential polygon is equilateral if and only if it is regular.[1]

All equilateral quadrilaterals are convex, but concave equilateral pentagons exist, as do concave equilateral polygons with any larger number of sides.

Viviani's theorem generalizes to equilateral polygons.[citation needed]

The principal diagonals of a hexagon each divide the hexagon into quadrilaterals. In any convex equilateral hexagon with common side a, there exists[2]:p.184,#286.3 a principal diagonal d1 such that

\frac{d_1}{a} \leq 2

and a principal diagonal d2 such that

\frac{d_2}{a} > \sqrt{3}.

Triambi, which are equilateral hexagons with trigonal symmetry,[3] appear in the three triambic icosahedra:

See also[edit]

References[edit]

  1. ^ De Villiers, Michael, "Equiangular cyclic and equilateral circumscribed polygons", Mathematical Gazette 95, March 2011, 102-107.
  2. ^ Inequalities proposed in “Crux Mathematicorum”, [1].
  3. ^ Dice of the Dimensions

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