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From: leeper@ahutb.UUCP (m.r.leeper)
Newsgroups: net.puzzle,net.jokes
Subject: Re: Re: Manhole covers (interview question)
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Date: Thu, 21-Mar-85 21:54:53 EST
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Posted: Thu Mar 21 21:54:53 1985
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On whether a Rouloux triangle can really cut I square hole as I
previously claimed.

 >Nope.  Apparantly, the Rouloux drill bit has 90 degree
 >corners.  This is just the right angle for fitting in the
 >corner of a square;  any other angle (such as 60 degrees, as
 >you suggest) would make the drill bit hit the corner of the
 >frame with a lot of force; the contraption would wear down
 >quickly.  ---

That was my original reasoning.  I still believe it to be the way it
was described to me.  Just after I published my original statement I
did the geometry and discovered to my embarrassment that the true
Rouloux triangle has 120 degree corners.  Unless my geometry is wrong.
Let me give my argument, I could be making an error.

To construct a Rouloux triangle start with an equilateral triangle with
vertices A, B, and C.  Now draw the short arc from A to B with center
C, from A to C with center B, and from B to C with center A.  Now how
do we measure the vertex angle at B.  Let c be the line tangent to the
arc from A to B with center c.  Line segment BC must be perpendicular
to c since it is tangent to a circle with center c, namely the
continuation of the arc from A to B.  Angle ABC is 60 degrees, since it
is a corner of an equilateral triangle.  This makes the acute angle 
formed by segment AB and c to be 30 degrees.  Hence the whole vertex
angle at B is 30 + 60 + 30 degrees.  That's 120 degrees.

				Mark Leeper
				...ihnp4!ahutb!leeper