Equilateral polygon

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 widths. All equilateral triangles are similar to each other, and have 60 degree internal angles.

An equilateral quadrilateral is a rhombus, described by a single angle α, which includes the square as a special case.

A convex equilateral pentagon can be described by two angles α and β with α ≥ β provided the fourth angle (δ) is the smallest of the rest of the angles.

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.

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.

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

${\frac {d_{1}}{a}}\leq 2$

and a principal diagonal d₂ such that

${\frac {d_{2}}{a}}>{\sqrt {3}}.$

Triambi

Triambi, which are equilateral hexagons with trigonal symmetry:

• 
  Concave
• 
  Self-intersecting

External links

• Equilateral triangle With interactive animation
• A Property of Equiangular Polygons: What Is It About? a discussion of Viviani's theorem at Cut-the-knot.