----- Original Message -----
Sent: Wednesday, July 30, 2003 4:16
PM
Subject: Re: [math-fun] Heron,
Brahmagupta, Robbins, etc.
Henry writes:
<<
In yesterday's (Tue, Jul
29) Wall Street Journal (!), a front
page article talked about David P.
Robbins's search for a generalization
of Heron's & Brahmagupta's
formulae for the area of polygons inscribed
in circles, when given only
their edge lengths.
The article indicated that Dr. Robbins was using
only pencil & paper,
which I found a bit odd, considering that there
are a number of symbolic
algebra systems that I would imagine would be
quite useful for this
problem.
>>
I wasn't able to access
the article, but given the above information I don't see what the puzzle
is. As long as a polygon inscribed in a circle is non-self-intersecting
and contains the center of the circle in its interior, its area is the sum of
the areas of all triangles T_E formed by each edge E and the center of the
circle.
Assuming radius = 1, and that E = edgelength, the area A_E of
T_E is given by
A_E = E * sqrt( 1 - (E/2)^2 ) / 2.
(If the center of
the circle is not in the interior of the polygon and/or the polygon is
self-intersecting, then the polygon's area is a signed sum of these triangles'
areas.)
What am I missing?
--Dan
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