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 solution for part 1 by Charlie)

Changed

For h = 1 To tot / 2

to

For h = 1 To tot -1

to allow for heights that are greater than the width, producing the following additional values, when the total of the height plus width does not exceed 2000.  However, in these cases AE exceeds AB, so the rhombus goes outside the bounds of the rectangle.  The previous solution stands.

width height
  AB    BC   AE  AC   EF   width*height width/height
  242 1320 3721 1342 7320    319440     .183333333333333
  162  720 1681  738 3280    116640     .225            
   98  336  625  350 1200     32928     .291666666666667
  288  840 1369  888 2590    241920     .342857142857143
   50  120  169  130  312      6000     .416666666666667
  128  240  289  272  510     30720     .533333333333333
   18   24   25   30   40       432     .75             
  800  840  841 1160 1218    672000     .952380952380952

Again, these are not real solutions, but attempts to come up with solutions where the width is less than the height.

Posted by Charlie on 2008-04-14 15:53:39