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 aided. by brianjn)

In looking at Charlie's data I did feel a little unnerved at first until I realised how we had approached the problem.

My values "a" + "c" = AB, or more precisely "a"+"d", but "c" = "d" anyway.

What might be noted when looking down my "a" column under "Partial Result Listing" is that the multiple of 7 suddenly goes awry, but that is not to be unexpected when one has a multiplicity of creating Pythagorean triangles.  Naturally that suggests that there are other solutions for which a Pythagorean triple is lowest valued base for a set of related figures.

Posted by brianjn on 2008-04-15 01:22:10