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 Here it is by FrankM)

For

k = 6, m = 20, n = 15, h = 37, j = 13  for case 1

k = 15, m = 60, n = 52, h = 113, j = 17  for case 2.

that is

AD = BC = 6, BE = 20, EA = 15, AC = 37, EF = 13  for case 1

AD = BC = 15, BE = 60, EA = 52, AC = 113, EF = 17  for case 2.

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.

 

Posted by Charlie on 2008-04-14 14:16:35