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.

Given that AECF is a rhombus we can note that
AE = AF = CE = CF.
Given that ABCD is a rectangle we can note that
AD = BC and AB = CD.
We can also note that BE = FD; and, if we draw a line segment perpendicular to AB from E to a point G on CD, we can note that EG = AD = BC and CG = BE = FD.

Let us assign simple variables to the lengths of the sides of the line segments:
a => AD = BC = EG
b => BE = FD = CG
c => AE = AF = CE = CF

We can note that...
AF is the hypotenuse of length c of right triangle ADF with legs of lengths b and c;
EF is the hypotenuse of right triangle EGF, with legs of lengths a and (c - b); and,
AC is the hypotenuse of right triangle ADC with legs of lengths a and (c + b).

Thus, we are looking for three Pythagorean triplets with a leg of a common length a and where SQRT(a² + (c - b)²) and SQRT(a² + (c + b)²) are also integers.

With the assistance of Euclid's formula, we can find that the smallest solution for the three Pythagorean triplets as
(7, 24, 25), (18, 24, 30), (24, 32, 40), which we can also derive the line segment lengths as follows:
AB = 32; BC = 24; AE = 25; AC = 40; and EF = 30
Therefore the rectangle has side lengths of 32 and 24 and diagonal length of 40; and the rhombus has a side length of 25, long diagonal length of 40 and short diagonal length of 30.

Posted by Dej Mar on 2008-04-15 10:38:20