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The Devil 30-gon (Posted on 2018-04-17) Difficulty: 3 of 5
If A1, A2, A3, . . . , A30 are the vertices of a regular 30-gon inscribed in a unit circle, then find

|A1A2|2 + |A1A3|2 + |A1A4|2 + ... + |A1A30|2

No Solution Yet Submitted by Danish Ahmed Khan    
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Solution computer-aided solution | Comment 1 of 3
DefDbl A-Z
Dim crlf$, dr


Private Sub Form_Load()
 Form1.Visible = True
 Text1.Text = ""
 crlf = Chr(13) + Chr(10)
 
 pi = Atn(1) * 4
 dr = pi / 180
 
 cAngle1 = 360 / 30
 
 For v2 = 2 To 30
   cAngle = (v2 - 1) * cAngle1
  ' If cAngle > 180 Then cAngle = 180 - cAngle
   halfAngle = cAngle / 2
   length = 2 * Sin(halfAngle * dr)
   tot = tot + length * length
 Next
 
 Text1.Text = Text1.Text & tot & crlf & " done"
  
End Sub

finds the sum of the squares of the chords from one point to be 60.

FWIW, that makes the total for all the chords 60*30/2 = 900, as each of the 30 vertices shares each of its 60 chords with another vertex and so would be counted twice if not divided by 2.

  Posted by Charlie on 2018-04-17 11:38:15
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