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Geometric calculator (Posted on 2006-11-24) Difficulty: 3 of 5
Given three segments of length 1, a and b in the plane, how can one construct segments of length a+b, |a-b|, ab, a/b, √a using ruler and compass? Which other calculator functions can be performed by geometric construction?

See The Solution Submitted by JLo    
Rating: 4.0000 (1 votes)

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Solution Solution | Comment 1 of 3

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| = |a-b|.
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,
          |AC||BD|
  |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 2006-11-24 10:42:24
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