Re: Maths Area Problem

On Sep 29, 11:02 am, "Philippe 92" <nos...@xxxxxxxxxxxx> wrote:

The maximum area is when the quadrilateral becomes cyclic.
Then the area is given by Brahmagupta formula, generalizing Heron's.

A = sqrt((s-a)*(s-b)*(s-c)*(s-d))

A sci.math thread I read recently gave (essentially) this elegant
proof of the fact that the maximal area of any polygon with prescribed
side lengths is cyclic:

If the side lengths determine a polygon at all, they determine a
cyclic one. Now affix each segment of the circumscribed circle to the
sides, and consider the total area of the polygon plus the affixed
segments as you vary the angles between edges. The configuration of
maximal area is the circle, because the circle has maximal area of any
shape with this fixed perimeter. Because the sum of the segment areas
is fixed, it follows that the polygon must also attain maximal area in
this configuration.

.

Relevant Pages

  • ANN: FastGEO Computational Geometry Library
    ... * 2D/3D Segment intersection point calculation ... * 2D/3D Triangle, quadix, rectangle, circle and polygon perimeter ... * Closest point on segment from a point ...
    (borland.public.delphi.thirdpartytools.general)
  • Re: Polygons of large area
    ... edges of the n-gon have equal length, which means the n-gon is regular. ... The WWW gives a good argument that the polygon is equilateral, ... remains to show that the equilateral n-gon of maximal area is regular. ... Now affix each segment of the circumscribed circle to the ...
    (sci.math)
  • Re: Polygons of large area
    ... edges of the n-gon have equal length, which means the n-gon is regular. ... The WWW gives a good argument that the polygon is equilateral, ... remains to show that the equilateral n-gon of maximal area is regular. ... Now affix each segment of the circumscribed circle to the ...
    (sci.math)
  • Re: Max Area for Figure Again
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    (sci.math)
  • Re: Maths Area Problem
    ... If the side lengths determine a polygon at all, ... Now affix each segment of the circumscribed circle to ... maximal area is the circle, because the circle has maximal area of ...
    (sci.math)
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