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Cover the Circumference (Posted on 2010-05-25) Difficulty: 4 of 5
Start with a unit circle and draw four line segments each of length L. The first line is a chord: it begins and ends on the circle. Subsequent lines start from the midpoint of the previous line and end on the circle. If the fourth line ends where the first line begins, what is L?

Note that this sketch is an approximation.

No Solution Yet Submitted by Larry    
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Some Thoughts Another approach Comment 4 of 4 |

I am trying a different approach and setting the internal quadrilateral AHIJ on a coordinate grid. I have
I(2-t, sqrt[1-t^2])
H({2-t-sqrt[(4t-1)/(5-4t)]*sqrt[1-t^2]}/2, {sqrt[1-t^2]-sqrt[(4t-1)/(5-4t)]*(2-t)}/2)

Then I extend AH to B, HI to C, and IJ to D by using the midpoint formula.  After that I wrote equations for the perpendictular bisectors of AB, AC, and AD.  All three lines must intersectinthe same point.  (I am omitting the details.)

I could not get any further analytically, simply from the complexity of the system.  But I was able to run a numeric simulation in UBASIC:

   10   Tlow=0.3
   11   Thigh=0.5
   12   Count=0
   20   T=(Tlow+Thigh)/2
   21   Count=Count+1
  101   K=sqrt((4*T-1)/(5-4*T))
  111   Ix=2-T
  112   Iy=sqrt(1-T*T)
  121   Hx=(Ix-K*Iy)/2
  122   Hy=(Iy+K*Ix)/2
  125   ' print Hx;Hy;Ix;Iy
  131   Bx=2*Hx
  132   By=2*Hy
  141   Cx=2*Ix-Hx
  142   Cy=2*Iy-Hy
  151   Dx=T+2
  152   Dy=-Iy
  160   ' print Bx;By,Cx;Cy,Dx;Dy
  200   Denom=2*Bx*Dy-2*By*Dx
  201   Xnum=Dy*(Bx*Bx+By*By)-By*(Dx*Dx+Dy*Dy)
  202   Ynum=Bx*(Dx*Dx+Dy*Dy)-Dx*(Bx*Bx+By*By)
  211   X=Xnum/Denom
  212   Y=Ynum/Denom
  220   Radius=sqrt(X*X+Y*Y)
  230   Delta=2*Cx*X+2*Cy*Y-Cx*Cx-Cy*Cy
  300   print T,Radius,Delta:print
  310   if Delta>0 then Thigh=T else Tlow=T
  320   if Count<40 then 20

This converges on a radius of 1.446685313104.  This corresponds to a chord length of 1.382470660263 when the circle has a unit radius.  This agrees with other numeric simulations.

  Posted by Brian Smith on 2010-05-27 00:27:57
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