A rectangle ABCD is circumscribed around a
rhombus AECF. The long sides of the rectangle coincide with two sides of the rhombus. Also, the rhombus and the rectangle share a common diagonal AC.
B E A
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C F D
What are the smallest dimensions when all the lengths AB, BC, AE, AC and EF are integers?
Find a parameterization of all such integral rectangle/rhombus pairs.
(In reply to
computer solution for part 1 by Charlie)
Charlie wrote:
Many results are found. The first few are:
AB = 32, BC = 24, AE = 25, EF =30
Question: doesn't this solution show the same fault as Charlie pointed out in my own contribution, namely that CE <> CF, so that AECF fails to be a rhombus (it is a parallelogram)
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BTW I reworked my earlier result to include the constraint CE = CF, or, equivalently: (nm)(n+m) = k^2. This gave (for both cases)
([p1 p2]^2  1) (p1^2  p2^2) = [2 p1 p2]^2
This equation is easily seen to have no solution.
At this point, the situation looks a bit like a logic problem. There are 3 characters: Charlie, BrianS and FrankM; and I'm beginning to think that at least two of them are missing something!
Edited on April 15, 2008, 12:37 am

Posted by FrankM
on 20080415 00:30:15 