DefDbl A-Z

Dim crlf$

Private Sub Form_Load()

 Form1.Visible = True

 

 

 Text1.Text = ""

 crlf = Chr$(13) + Chr$(10)

 

 For a = 1 To 9

  For b = 0 To a - 1

    ab = 10 * a + b

    ba = 10 * b + a

    sq = ab * ab - ba * ba

    sr = Int(Sqr(sq) + 0.5)

    If sr * sr = sq Then

      Text1.Text = Text1.Text & ab & Str(ba) & "     " & sq & Str(sr) & crlf

    End If

    DoEvents

  Next

 Next

 

 

 Text1.Text = Text1.Text & crlf & " done"

  

End Sub

finds

65 56     1089 33

giving the answer as PQ has length 65. The chord RS has length 56 and the length of OT is 33/2.

This is the result of segment OR (or OS) being a radius of the circle (half the length of PQ) and the hypotenuse of a right triangle with legs being to TO and half the length of RS. The halvings require the halving of the 33 result.

The important consideration is that in working with the Pythagorean result sqrt(ab^2/4 - ba^2/4) (the ab and ba being concatenations, not multiplications), the difference of the numerators is an integer which must be a square as no multiplication by a square will turn a non-square into a square or the square of a rational.