This is probably not a 'proof' in the strict sense ...
Assuming (x) and (y) being of different size and not unit:
Looking at (x*y) geometrically, the product gets interpreted as a
rectangle of width (y) and height (x); one will be able to construct
a square -similar in area- with side length (a) and area (a^2) by
taking the geometric mean a = sqrt(x*y).
The area measure will be totally arbitrary without a reference of
side (a) to a unit length.
To make the transition from area to corresponding length one needs to
count: so much amount of area (some number) translates to the same
amount (same number) of of length. And for counting, one needs a
unit; so many units of area translate to the same number of units of length.
-M
At 2016-07-14 17:19, you wrote:
Given line segments x and y, construct (using compass and straight edge)
line segments having the following values:
x+y
x-y
x*y
x%y
The first two are immediate. I have proven that x%y is impossible if you
are not given a reference length 1 (or some other reference length from
which to construct 1).
The problem is, prove or disprove that you construct x*y without using 1.
(Constructing x*y and x%y _with_ 1 are pretty easy.)
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