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Circle on a grid 1 (Posted on 2013-05-24) Difficulty: 3 of 5
Draw a unit circle on a sheet of graph paper with lines spaced 1 unit such that one square is completely inside the circle.

The lines of the paper will intersect the circle in 8 places, cutting the circle into 8 arcs. Consider the 4 arcs where some of the grid lines cut the circle as shown in the diagram: AB, BC, CD, DA.

Describe, as precisely as possible, how to place the circle so the relative lengths of these arcs are in the extended ratio 1:2:4:3 in order around the circle. Use the coordinate system as in the diagram.

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Not obvious to me | Comment 1 of 3
I am sure that this is possible with some radius R, but I understand the problem to ask for a unit circle (R = 1), and it is not at all obvious that it works with R = 1. Here is my concern:

Let O be the center of the circle.

In order for the arcs to to in ratio 1:2:4:3, we need
  Angle BOA = 36 degrees
  Angle AOD = 72 degrees
  Angle DOC = 144 degrees
  Angle COB = 108 degrees

Then lengths of line segments can be calculated
  length(AB) = 2*sin(18 degrees)
  length(BC) = 2*sin(54 degrees)
  length(AC) = 2*sin(72 degrees)
So ABC is a triangle with known sides.

There is only one way to place triangle ABC so that AC is on the X axis and B is on the Y axis.  

And then the location of O is forced.  It is where the perpendicular bisectors intercept.  Or distance 1 from any two of A,B, or C.  (Take your pick).

And once O is determined, then D is forced, as there is only one point on the Y axis other than B than is 1 unit from O.  But it is not obvious to me that angle DOC will in fact = 144 degrees.  I don't off hand see a reason that it should.  Perhaps somebody can prove this one way or another using Geometer's sketchpad?

As I said at the start, I am sure that this is possible for some radius R, but it don't offhand see that this can be made to work if R = 1.

  Posted by Steve Herman on 2013-05-24 22:56:45
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