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 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. -- __ __ __ __ __ ___ _____________________________________________ |__||__)/ __/ \|\ ||_ | / | || \\__/\__/| \||__ | /...Internet access for all Acorn RISC machines ___________________________/ paulg@argonet.co.uk