Editing Equilateral polygon (section)

==Measurement==
[[Viviani's theorem]] generalizes to equilateral polygons:<ref>{{citation|last=De Villiers|first=Michael|title=An illustration of the explanatory and discovery functions of proof|journal=[[Leonardo (journal)|Leonardo]]|year=2012|volume=33|issue=3|pages=1–8|doi=10.4102/pythagoras.v33i3.193|url=http://www.pythagoras.org.za/index.php/pythagoras/article/view/193/228|quote=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|doi-access=free}}.</ref> 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 ''d''<sub>1</sub> such that<ref name=Crux>''Inequalities proposed in “[[Crux Mathematicorum]]”'', [http://www.imomath.com/othercomp/Journ/ineq.pdf], p.184,#286.3.</ref>

:<math>\frac{d_1}{a} \leq 2</math>

and a principal diagonal ''d''<sub>2</sub> such that

:<math>\frac{d_2}{a} > \sqrt{3}</math>.