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### 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

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/
```
Associated Topics:
High School Geometry
High School Triangles and Other Polygons

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