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Topic: Equql area rectangles
Replies: 6   Last Post: Jan 20, 2004 5:34 PM

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John Conway

Posts: 2,238
Registered: 12/3/04
Re: Equql area rectangles
Posted: Jan 19, 2004 12:47 PM
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On 16 Jan 2004, Nadel Oxles wrote:

> Starting with a rectangle of random dimentions, using geometric
> construction only, consrtuct a second rectngle with the same area and
> the sides of a defined ratio.
>
> Example, you may start with a long narrow rectangle and then have to
> construct an equal area rectangle with sides in the ratio of 2:3
>
> I have been playing with this for a while and am beginning to think it
> is impossible. I can start with a random rectangle and construct a
> second rectangle with equal area and a given side, but not this
> problem.
>
> I look to the wisdom of this group for some guidance!


Here's a way to do it when the shape of the desired new rectangle
doesn't differ too much from that of the original one. Join A to
a point E on CD, and suppose the perpendicular from B to AE
hits that line at a point F between A and E.

A---------------B
|\ |\
| \ | \
| F | F'
D---E-----------C---E'
\
\
F"

Now define points E', F' by the conditions that EE' and FF'
be parallel and have the same length that AB, and F" by the
condition that AE and FF" be parallel and have the same length.
Then ABCD can be dissected to FBE'F".

You can change the shape arbitrarily much by repeatedly making such
"small" shape-changes.

John Conway





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