From: jrr@atml.co.uk (John Rickard)
Newsgroups: sci.math
Subject: Re: Q: Constructing the Geometric Mean
Date: 29 Jun 1997 14:24:45 +0100 (BST)

I wrote:
: : can the geometric mean of AB and AC be constructed _on_THIS_segment_
: : using "only a straightedge" (i.e., no compass)?
:
: No, because there are no two points D and E on the segment such that
: DE is the geometric mean of AB and AC and remains so under all
: projective transformations.  (Easy to prove.)

Or one can prove the impossibility of constructing the geometric mean
by noting that, using an unmarked straightedge, from points with
rational coordinates one can only construct points with rational
coordinates.

: [I'm assuming an unmarked straightedge.]

-- 
John Rickard

==============================================================================

From: Paul Gailiunas <paulg@argonet.co.uk>
Newsgroups: sci.math
Subject: Re: Q: Constructing the Geometric Mean
Date: Fri, 27 Jun 1997 22:01:34 GMT

In article <5ooesa$8f5@newstand.syr.edu>, rbsalgad@gamera.syr.edu (Rob
Salgado) wrote:
> 
> Given three points on the line segment
> 
>         A-------B---------------------------------C
> 
> can the geometric mean of AB and AC be constructed _on_THIS_segment_
> using "only a straightedge" (i.e., no compass)?
> If not, how about "with a straightedge and compass"?
> 
> rob
> salgado@physics.syr.edu
> 
With a straight edge and compass: Euclid Book III, Proposition 36 (if you're
interested) . Construct the circle with BC as diameter by finding the
mid-point of BC=M. The tangent from A to the circle is the required line
segment.
To get the tangent construct the circle with AM as diameter. The
intersection of the two circles gives the point of contact of the tangent
with the circle with BC as diameter (angle in a semicircle is a right
angle).

Mascheroni proved that any straight-edge and compass construction can be
done using only a compass.  Steiner and Poncelet proved that a straight edge and one circle is
sufficient but I have no idea how to solve your problem with this
restriction. Two references that look useful are 
Steiner J., Geometrical Constructions with a Ruler Given a Fixed Circle and
Its Center. (Trans. by Marion Stark, ed. Archibald R.C., New York: Yeshiva
University.) Scripta Mathematica 1950. 88
Tucker C., Construction for mean proportional. Mathematical Gazette
14:542-44 1929.


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