Polygons and TrianglesDate: 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/ |
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