Polygons and Triangles

Date: 03/09/99 at 16:42:14
From: Jonathan Randall
Subject: Polygons and Triangles

I have used loci to prove that isosceles triangles have a greater area 
than scalene triangles, when a regular n-sided polygon is split up 
into n triangles, with a set perimeter. But this does not specifically 
show that regular polygons have greater areas than irregular polygons.

Can you help?

Date: 03/11/99 at 12:01:00
From: Doctor Peterson
Subject: Re: Polygons and Triangles

You are on the right track. The approach I have in mind uses what you 
have proved, but requires a slight twist in your thinking, more or 
less a proof by contradiction.

I would take any non-regular polygon and show that you can find a 
larger polygon with the same perimeter. Then the largest polygon with 
that perimeter must be regular.

Suppose that there are three consecutive vertices A, B, and C in a 
polygon such that AB and BC have different lengths. See if you can 
find a point B' for which AB' and B'C are the same length, but their 
sum is the same as AB + BC. Then show that the area of triangle AB'C 
will be larger than that of ABC. (This is basically what you have told 
me already.) If you replace B with B' in the polygon, its perimeter 
stays the same but the area is larger.

                       B'
             B         +
             +       /   \
            /   \  /       \
           /     /  \        \
          /    /        \      \
         /   /             \     \
        /  /                   \   \
       / /                        \  \
    A //                              \\  C
     + - - - - - - - - - - - - - - - - - +
     |                                    \
     |                                     \
     ...                                    ...

This will show that the largest polygon has to have all sides the 
same, since a polygon whose sides are not the same is never the 
largest.

You will also have to show that the angles in the largest polygon with 
a given perimeter are all the same. Try a method similar to what we 
just did for the sides. (I have not taken the time to work that part 
out.)

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/