124 Basic geometric constructions
4.1 Geometric mean ..................
We present two ruler-and-compass constructions of the geometric means
of two quantities given as lengths of segments. These are based on Eu-
clid’s proof of the Pythagorean theorem.
Construction 4.1.Given two segments of lengtha<b, mark three
pointsP,A,Bon a line such thatPA= a,PB= b, andA,Bare
on thesameside ofP. Describe a semicircle withPBas diameter,
and let the perpendicular throughAintersect the semicircle atQ. Then
PQ^2 =PA·PB, so that the length ofPQis the geometric mean ofa
andb.
P A BQConstruction 4.2.Given two segments of lengtha,b, mark three points
A,P,Bon a line (PbetweenAandB) such thatPA=a,PB=b.
Describe a semicircle withABas diameter, and let the perpendicular
throughPintersect the semicircle atQ. ThenPQ^2 =PA·PB, so that
the length ofPQis the geometric mean ofaandb.
A P BQ