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.

I have redrawn the graphic with my own labels:

B          E                   A
+----a----+--------d-----+
|           /  º           .    / |
|         /     º       e     /  |
|       /        º  .        /    |
b     c         .º         /     |
|    /      .      º      /       |
|  /    .            f   /        |
|/ .                 º /          |
+---------+--g--+--------+
C           X          F       D        

Using the Pythagorean triangle [BEC] as my reference, with appropriately chosen dimensions, all designated lengths in the diagram are integers. One might note the presence of "b" as a common length in Δ BEC, Δ ABC and Δ EFX.

Now, because "c" is expressible in terms of "a" and "b", and "d"="c", then all measurements (other than "a" and "b") can be expressed in terms of "a" and "b".

My list, and I have taken into account multiples of each possible solution with the factor "n" being from 1 to ∞, is:
a = n * a
b = n * b
c = n * √(a²+b²)
d = n * √(a²+b²); I have been careful not to equate "d" to "n * c"
e = n * √((√(a² + b²) +a)² +b²)
f =  n * √(b² + (√(a² + b²)-a)² )
g = n * (√(a² + b²)-a)


In a spreadsheet I generated the following;
n  a   b  c   d   e   f   g
1  7 24 25 25 40 30 18

The values "a" and "b" were located respectively in A1 and B1 with the values of "a" and "b" being replaced with "$A$1" and "$B$1" in the above formulae.  Copied into columns headed as above and allowing the "n" value to increment, one can copy rows down to produce related values.
I used the "Drawing" Toolbar in Excel to recreate the above drawing using cells underlaid and close by to reflect values of the first line of a newly created table for which some such values follow.

On changing the (a,b) values to (41,840), (119,120), (161, 240) and (570,336) I had these as valid solutions.
Those values were derived from a computer program listed below.  The listing does not use "a" and "b" as sole parameters for the values mentioned above.

Program Listing
Note: "a" and "b" have been given upper limits of 1000 each.

          f$ = "pythag.txt"
CLS
     OPEN f$ FOR OUTPUT AS #1

 FOR a = 1 TO 1000
 FOR b = 1 TO 1000

     c = SQR(a * a + b * b)
     d = c
     e = SQR(b * b + (a + c) * (a + c))
     g = d - a
     f = SQR(b * b + (c - a) * (c - a))
     
     IF (e = INT(e) AND f = INT(f)) THEN
        PRINT a; b; c; d; e; f; g
        a$ = STR$(a) + CHR$(44) + STR$(b) + CHR$(44) + STR$(c) + CHR$(44) + STR$(d) + CHR$(44) + STR$(e) + CHR$(44) + STR$(f)+CHR$(44)+STR$(g)

        PRINT #1, a$
     END IF

  NEXT
  NEXT
    CLOSE 1
   
Partial Result Listing
  a   b    c   d    e   f    g
  7, 24, 25, 25, 40, 30, 18
 14, 48, 50, 50, 80, 60, 36
 21, 72, 75, 75, 120, 90, 54
 28, 96, 100, 100, 160, 120, 72
 35, 120, 125, 125, 200, 150, 90
 41, 840, 841, 841, 1218, 1160, 800
 42, 144, 150, 150, 240, 180, 108
 49, 168, 175, 175, 280, 210, 126
 56, 192, 200, 200, 320, 240, 144
 63, 216, 225, 225, 360, 270, 162
 70, 240, 250, 250, 400, 300, 180
 77, 264, 275, 275, 440, 330, 198
 84, 288, 300, 300, 480, 360, 216
 91, 312, 325, 325, 520, 390, 234
 98, 336, 350, 350, 560, 420, 252
 105, 360, 375, 375, 600, 450, 270

Edited on April 15, 2008, 12:15 am

Posted by brianjn on 2008-04-14 22:24:19