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Minimizing Perimeter (Posted on 2015-03-04) Difficulty: 2 of 5
Let ABC be a triangle with integral side lengths such that angle A=3 * angle B. Find the minimum value of its perimeter.

No Solution Yet Submitted by Danish Ahmed Khan    
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Solution computer solution | Comment 3 of 15 |
The minimum value of the perimeter is 21 (10:8:3 triangle):

DefDbl A-Z
Dim fct(20, 1), crlf$

Private Sub Form_Load()
 Text1.Text = ""
 crlf$ = Chr(13) + Chr(10)
 Form1.Visible = True
 
  For peri = 3 To 100
    For a = Int(peri / 2) To peri / 4 Step -1
      For b = 1 To a - 1
        c = peri - a - b
        If a + b > c And a + c > b And b + c > a Then
           cosa = (-a * a + b * b + c * c) / (2 * b * c)
           If Abs(cosa) < 1 Then
           anglea = acos(cosa)
           cosb = (-b * b + a * a + c * c) / (2 * a * c)
           If Abs(cosb) < 1 Then
           angleb = acos(cosb)
           rat = anglea / angleb: ratround = Int(rat + 0.5)
           If Abs(ratround - rat) < 0.000001 And ratround = 3 Then
             Text1.Text = Text1.Text & a & Str(b) & Str(c) & "    " & peri & Str(rat)
             Text1.Text = Text1.Text & "    " & anglea * 180 / (4 * Atn(1)) & Str(angleb * 180 / (4 * Atn(1))) & crlf
             DoEvents
           End If
           End If
           End If
        End If
      Next
    Next
  Next
 

  Text1.Text = Text1.Text & "done"
  DoEvents

End Sub
Function acos(x)
  If x = 0 Then acos = 2 * Atn(1): Exit Function
  ac = Atn(Sqr(1 - x * x) / x)
  If ac > 0 Then acos = ac Else acos = ac + 4 * Atn(1)
End Function


finds

10 8 3    21 3    124.228866327813   41.4096221092709
20 16 6    42 3    124.228866327813   41.4096221092709
30 24 9    63 3    124.228866327813   41.4096221092709
40 32 12    84 3    124.228866327813   41.4096221092709

showing for each: three sides, a, b and c; perimeter; ratio of angles; and the measures of the angles in degrees.

  Posted by Charlie on 2015-03-04 09:33:20
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