# Maximum ratio of the area of two regions?

A regular hexagon is split into two regions by a straight line so that the ratio of the perimeters of these regions is 2:1. Find the maximum ratio of the areas of the two regions.

This problem is inspired by one I saw on Brilliant.org today (credit goes to Digvijay Singh). It went as follows,

An equilateral triangle is split into two regions with equal perimeters. Find the maximum ratio of the areas of the two regions.

The region to max/minimise is a quadrilateral, composed of a red triangle with base $$x$$ and altitude $${\sqrt3\over2}$$, and a blue triangle with base $$1-x$$ and altitude $${\sqrt3\over2}(1+x)$$ (see picture). The area of the region is therefore $$A={\sqrt3\over4}(1+x-x^2),$$ which attains its maximum for $$x=1/2$$ and its minimum for $$x=0$$ or $$x=1$$.