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From: jml@cs.strath.ac.uk (Joseph McLean)
Newsgroups: sci.math
Subject: pentagons within squares
Message-ID: <303@stracs.cs.strath.ac.uk>
Date: Tue, 4-Nov-86 07:22:25 EST
Article-I.D.: stracs.303
Posted: Tue Nov  4 07:22:25 1986
Date-Received: Fri, 7-Nov-86 22:00:27 EST
Reply-To: jml@cs.strath.ac.uk (Joseph McLean)
Organization: Department of Computer Science at Strathclyde University, UK.
Lines: 22


I think Dave desJardins has hit the nail on the head.The largest pentagon
inscribed in a square is larger than the largest pentagon inscribed in
the largest circle inscribed in the square.

(How about that for a mouthful).

Simply conceive of placing one of the corners of the circled-pentagon 
at the middle of one of the square's sides.Since the distance from the
centre of this pentagon to the corner is the radius of the circle(which
is half the length of a side of the square),and this distance is the
smallest distance from the centre of the square to one of its sides
(varying from r to sqrt(2)*r),the other 4 corners of the pentagon will
not touch the square.Hence if we magnify the pentagon from the point in
contact with the square,we eventually obtain a strictly larger one
that touches the square in 3 places.Hence result.

What we can say is that the largest polygon inscribable inside a square
is "at least as large as" the largest polygon inscribable inside the
largest circle inscribable inside the square.

     jml,the circumlocutive mathematician.