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A rectangle Around A rhombus (Posted on 2008-04-14) Difficulty: 4 of 5
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.

No Solution Yet Submitted by Brian Smith    
Rating: 3.5000 (2 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
re: Here it is | Comment 3 of 13 |
(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
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