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
+-------+---------------+
|      /               /|
|     /               / |
|    /               /  |
|   /               /   |
|  /               /    |
| /               /     |
|/               /      |
+---------------+-------+
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)

____________________

BTW I reworked my earlier result to include the constraint CE = CF, or, equivalently: (n-m)(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 2008-04-15 00:30:15