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.

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.