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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)

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Solution computer solution for part 1 | Comment 5 of 13 |

Private Sub Form_Load()
  Open "rectangle about rhombus 2.txt" For Output As #2
  FontTransparent = False
  For tot = 7 To 6000
   CurrentX = 1: CurrentY = 1
   Print tot;
   For h = 1 To tot / 2
    Visible = True
    DoEvents
    l = tot - h
    diag1 = Int(Sqr(h * h + l * l) + 0.5)
    If diag1 * diag1 = h * h + l * l Then
      topR = Int(diag1 * diag1 / (2 * l) + 0.5)
      If topR = diag1 * diag1 / (2 * l) Then
        legA = 2 * topR - l
        diag2 = Int(Sqr(legA * legA + h * h) + 0.5)
        rhomSide = Sqr(h * h + (l - topR) * (l - topR))
        If diag2 * diag2 = legA * legA + h * h Then
         If rhomSide = topR Then
          pu 5, l: pu 5, h: pu 5, topR: pu 5, diag1: pu 5, diag2: pu 5, rhomSide
          pu 10, l * h: pu 20, l / h
          Print #2,
         End If
        End If
      End If
    End If
   Next
  Next
 Print
 Print "done"
 
 Close
End Sub
Sub pu(l, n)
  s$ = LTrim(Str(n))
  If Len(s$) < l Then
   s$ = Right(Space$(l) + s$, l)
  End If
  Print #2, s$;
End Sub

Many results are found. The first few are:

width height
  AB    BC   AE  AC   EF   width*height width/height
   32   24   25   40   30        768    1.33333333333333

   64   48   50   80   60       3072    1.33333333333333
   96   72   75  120   90       6912    1.33333333333333
  128   96  100  160  120      12288    1.33333333333333
  160  120  125  200  150      19200    1.33333333333333
  192  144  150  240  180      27648    1.33333333333333
  224  168  175  280  210      37632    1.33333333333333
  288  120  169  312  130      34560    2.4
  256  192  200  320  240      49152    1.33333333333333
  288  216  225  360  270      62208    1.33333333333333
  320  240  250  400  300      76800    1.33333333333333


Of the many others, the smallest for each ratio of width to height is found below, arranged in order of width/height ratio. Integral multiples of the dimensions result in the area being multiplied by the square of the linear multiple, but the width/height ratio remains the same. There are presumably more (besides just the multiples of these) if we allow the total of the height + width to exceed 6000. For a given ratio of width to height, all the other measurements are in proportion.

width height
  AB    BC   AE  AC   EF   width*height width/height
  882  840  841 1218 1160     740880    1.05           
   32   24   25   40   30        768    1.33333333333333
  450  240  289  510  272     108000    1.875          
  288  120  169  312  130      34560    2.4            
 2450  840 1369 2590  888    2058000    2.91666666666667
 1152  336  625 1200  350     387072    3.42857142857143
 3200  720 1681 3280  738    2304000    4.44444444444444

  Posted by Charlie on 2008-04-14 15:34:31
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