Equilateral polygon

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).

Thus if the number of sides n is odd, a tangential polygon is equilateral if and only if it is regular.

[1] 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.

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.