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 re: Here it is by Charlie)

Charlie wrote:

However, BE=20 and BC=6 make EC=sqrt(436), which is not equal to EA=15, and the purported rhombus is not a rhombus but a rhomboid.  Likewise with 60 and 20 not resulting in hypotenuse equal to 52.

FrankM responds:

Right you are! Obviously I forgot a constraint. I'll  go back and see what a needle, thread, and a plaid patch can do!

Posted by FrankM on 2008-04-14 23:16:08