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?
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).
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Posted by Bractals
on 2006-11-24 10:42:24 |
Solution
