Equilateral polygon

In geometry, an equilateral polygon is a polygon which has all sides of the same length. Except in the triangle case, an equilateral polygon does not need to also be equiangular (have all angles equal), but if it does then it is a regular polygon. If the number of sides is at least five, an equilateral polygon does not need to be a convex polygon: it could be concave or even self-intersecting.

Examples

All regular polygons and edge-transitive polygons are equilateral. When an equilateral polygon is non-crossing and cyclic (its vertices are on a circle) it must be regular. An equilateral quadrilateral must be convex; this polygon is a rhombus (possibly a square).

Convex equilateral pentagon
Concave equilateral pentagon

A convex equilateral pentagon can be described by two consecutive angles, which together determine the other angles. However, equilateral pentagons, and equilateral polygons with more than five sides, can also be concave, and if concave pentagons are allowed then two angles are no longer sufficient to determine the shape of the pentagon.

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]

Measurement

Viviani's theorem generalizes to equilateral polygons:[2] The sum of the perpendicular distances from an interior point to the sides of an equilateral polygon is independent of the location of the interior point.

The principal diagonals of a hexagon each divide the hexagon into quadrilaterals. In any convex equilateral hexagon with common side a, there exists a principal diagonal d1 such that[3]

and a principal diagonal d2 such that

.

Optimality
Main article: Reinhardt polygon
Four Reinhardt pentadecagons

When an equilateral polygon is inscribed in a Reuleaux polygon, it forms a Reinhardt polygon. Among all convex polygons with the same number of sides, these polygons have the largest possible perimeter for their diameter, the largest possible width for their diameter, and the largest possible width for their perimeter.[4]

References
  1. De Villiers, Michael (March 2011), "Equi-angled cyclic and equilateral circumscribed polygons" (PDF), Mathematical Gazette, 95: 102–107, doi:10.1017/S0025557200002461.
  2. De Villiers, Michael (2012), "An illustration of the explanatory and discovery functions of proof", Leonardo, 33 (3): 1–8, doi:10.4102/pythagoras.v33i3.193, explaining (proving) Viviani's theorem for an equilateral triangle by determining the area of the three triangles it is divided up into, and noticing the 'common factor' of the equal sides of these triangles as bases, may allow one to immediately see that the result generalises to any equilateral polygon.
  3. Inequalities proposed in “Crux Mathematicorum”, , p.184,#286.3.
  4. Hare, Kevin G.; Mossinghoff, Michael J. (2019), "Most Reinhardt polygons are sporadic", Geometriae Dedicata, 198: 1–18, arXiv:1405.5233, doi:10.1007/s10711-018-0326-5, MR 3933447, S2CID 119629098
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