Given three segments of length 1, a and b in the plane, how can one construct segments of length a+b, ab, ab, a/b, √a using ruler and compass? Which other calculator functions can be performed by geometric construction?
Construct concentric circles (radii a and b) with a
line through their center intersecting the circles at
points A, B, C, and D in that order. Then,
AC = BD = a+b and AB = CD = ab.
Construct rays ABD and ACE separated by an angle of
approximately 60 degrees such that DE is parallel to BC.
Triangles ABC and ADE are similar. Therefore,
AB AD AB + BD
 =  = 
AC AE AC + CE
Thus,
ACBD
CE = 
AB
Cases:
AC BD AB CE

a b 1 ab
a 1 b a/b
1 1 a 1/a
Construct a circle with diameter AB such that
AB = a+1. Construct a point C on diameter AB
such that AC = a. Construct a line through point
C perpendicular to AB intersecting the circle at
points D and E. Then,
CD = CE = sqrt(a).

Posted by Bractals
on 20061124 10:42:24 